General foundations of high-order immersed interface methods to solve interface problems

1 Introduction

High-order numerical solutions to differential equations arising from discontinuous solutions find extensive utility across various research domains [1,2,3,4,5,6]. In the case of smooth solutions, the standard central finite-difference method requires a significant number of grid points to achieve a high level of accuracy in its numerical results. As a result, over the past few decades, several schemes have been developed to obtain fourth- and sixth-order finite-difference methods [7,8,9,10,11], including those one based on the implicit finite-difference (IFD) formulation [12,13,14,15].

On the other hand, although several methods have been proposed to address discontinuous problems [2,16,17,18,19,20,21], the immersed interface method (IIM) [22,23,24,25] stands out as a highly accurate option that requires minimal adjustments to the standard finite-difference formulation. However, these methods typically achieve second-order accuracy. For instance, there are limited implementations of a few third-, fourth- and sixth-order IIMs available for solving Poisson equations with discontinuous solutions [26,27,28,29,30,31,32,33].

This paper focuses on the basic ideas of combining the IFD and IIM to achieve high-order approximations for second-order derivatives of both continuous and discontinuous real-valued functions. The IFD scheme offers a highly accurate numerical method [34,35], while the IIM handles discontinuities through minimal adjustments made exclusively at grid points where the stencil intersects the interface [36,37], yielding additional terms known as jump contributions. The resulting method will be named as implicit finite-difference immersed interface method (IFD-IIM).

We illustrate the implementation of a global fourth-order IFD-IIM approach with two examples: the one-dimensional Poisson equation for static cases and the heat conduction equation with a fixed interface for time-evolving scenarios. Our proposed method offers several advantages. Notably, the resulting tridiagonal matrix coefficient of the finite-difference scheme remains the same as those for smooth solutions, with the additional terms arising from the jumps located in the right-hand side vector. Consequently, our algorithm is straightforward to implement, employing the efficient Thomas' algorithm.

We have organized our study as follows. In Section 2, we introduce a fourth-order IFD method capable of handling second-order derivatives, both in smooth and discontinuous scenarios. Section 3 demonstrates the application of this implicit scheme in approximating solutions to the one-dimensional Poisson equation. Section 4 shows the combination of the IFD-IIM with the Crank-Nicolson method to solve the heat conduction equation. Sections 5 and 6 provide a series of numerical examples to illustrate the algorithm's accuracy for both equations. Lastly, Section 7 offers our conclusions and outlines directions for future research.

2 IFD formulation with discontinuities

In this section, we outline the key attributes of the IFD formulation to achieve a global fourth-order method, demonstrating how the scheme can be adapted for addressing discontinuous problems through the utilization of the IIM.

We approximate the numerical solution on the domain ${\textstyle [{\mathfrak {a}},\,{\mathfrak {b}}]}$ that is divided into ${\textstyle N}$ sub-intervals using the points ${\textstyle x_{i}}$, as follows

$x_{i}={\mathfrak {a}}+(i-1)h,\qquad i=1,2,\dots ,N,N+1,$

(1)

where the grid size is given by ${\textstyle h=({\mathfrak {b}}-{\mathfrak {a}})/N}$. We employ ${\textstyle U_{i}}$ and ${\textstyle u_{i}=u(x_{i})}$ to denote the approximate and exact solution at the ${\textstyle i}$th-point of the grid, respectively. Here, the interface is at ${\textstyle x=x_{\alpha }}$ located between grid points ${\textstyle x_{I}}$ and ${\textstyle x_{I+1}}$ as ${\textstyle x_{I}\leq x_{\alpha }{<}x_{I+1}}$, see Fig. 1. The distances of the closest grid points to the interface are defined as ${\textstyle h_{L}=x_{I}-x_{\alpha }}$ and ${\textstyle h_{R}=x_{I+1}-x_{\alpha }}$. Note that ${\textstyle h_{R}}$ and ${\textstyle h_{L}}$ are positive and negative values, respectively.

Figure 1: Example of a discontinuous function $u$ with an interface located between the points $x_{I}$ and $x_{I+1}$.

On the other hand, the jump for ${\textstyle u}$ at ${\textstyle x_{\alpha }}$ is defined as

$\left[u\right]=u^{+}-u^{-},\quad \mathrm {where} \quad u^{+}=u(x_{\alpha }^{+})=\lim _{x\to x_{\alpha }^{+}}u(x),\quad \mathrm {and} \quad u^{-}=u(x_{\alpha }^{-})=\lim _{x\to x_{\alpha }^{-}}u(x).$

We employ a similar definition for the jumps such as the ones of the right-hand side and the derivatives of ${\textstyle u}$.

In this paper, we designate ${\textstyle x_{I}}$ and ${\textstyle x_{I+1}}$ as irregular points, while considering the rest as regular points. This classification holds significant importance as distinct schemes are applied to each category, as presented in the following theorems.

2.1 Second-order derivative approximation for regular and irregular grid points

The following two Theorems state the main results to approximate the second-order derivative using high-order schemes for regular and irregular points.

Theorem 1: Regular points [34,35]. Let us consider a real-valued function ${\textstyle u}$ with an interface ${\textstyle x_{\alpha }}$ such that ${\textstyle x_{I}\leq x_{\alpha }{<}x_{I+1}}$. Then ${\textstyle u_{xx}}$ can be approximated by a fourth-order IFD scheme

$\left.{\mathfrak {D}}^{2}u_{xx}\right|_{i}=\left.\delta ^{2}u\right|_{i}+O(h^{4}),\quad i\neq I,I+1,$

(2)

where

${\mathfrak {D}}^{2}(\cdot ):=(\cdot )+bh^{2}(\cdot )_{xx},$

(3)

${\textstyle b=1/12}$, and the central finite-difference formula ${\textstyle \delta ^{2}}$ is given by

$\left.\delta ^{2}u\right|_{i}:={\frac {1}{h^{2}}}\left\{u_{i-1}-2u_{i}+u_{i+1}\right\}.$

(4)

Proof.

From Taylor series expansions and under some simplifications, the second-order derivative at any regular point ${\textstyle x_{i}}$ can be written in terms of the centered finite-difference operator, as follows

$\left.u_{xx}\right|_{i}={\frac {1}{h^{2}}}\left\{u_{i-1}-2u_{i}+u_{i+1}\right\}-{\frac {1}{12}}h^{2}\left.u_{xxxx}\right|_{i}+O(h^{4}),$

Previous equation holds due to the solution is smooth on a neighborhood around ${\textstyle x_{i}}$. Thus, we obtain a fourth-order IFD using a stencil of three nodes by moving the fourth-order term to the left-hand side. We get

$\left.u_{xx}\right|_{i}+{\frac {1}{12}}h^{2}\left(\left.u_{xx}\right)_{xx}\right|_{i}={\frac {1}{h^{2}}}\left\{u_{i-1}-2u_{i}+u_{i+1}\right\}+O(h^{4}).$

Finally, the proof is completed by using the definition of operator ${\textstyle {\mathfrak {D}}^{2}}$, ${\textstyle \delta ^{2}}$, and ${\textstyle b}$. This completes the proof. ♦

It is important to remark that formula (2) is applicable exclusively for regular points. In order to address this limitation for the two irregular points near the interface, we introduce a modified IFD scheme using the IIM, specifically tailored to handle discontinuous solutions. Furthermore, instead of having a fourth-order local truncation error for the irregular points, we proceed as other IIMs [22,23,24,29,33,32] by taking one order lower at these points. We will numerically show that the global order of convergence can be still ${\textstyle O(h^{4})}$ even if the local truncation error at ${\textstyle i=I}$ and ${\textstyle i=I+1}$ is ${\textstyle O(h^{3})}$.

Theorem 2: Irregular points [14]. Let us consider the known jump conditions

$\left[u\right],\quad \left[u_{x}\right],\quad \left[u_{xx}\right],\quad \left[u_{xxx}\right],\quad \left[u_{xxxx}\right],\quad$

(5)

at ${\textstyle x_{\alpha }}$ such that ${\textstyle x_{I}\leq x_{\alpha }{<}x_{I+1}}$. Then ${\textstyle u_{xx}}$ can be approximated at ${\textstyle x_{I}}$ and ${\textstyle x_{I+1}}$ by the IFD-IIM given by

${\mathfrak {D}}^{2}u_{xx}\left.\right|_{i}=\left.\delta ^{2}u\right|_{i}+C_{i}+O(h^{3}),\quad i=I,I+1,$

(6)

where ${\textstyle {\mathfrak {D}}^{2}}$ and ${\textstyle \delta ^{2}}$ are defined in (3) and (4), respectively, and

$C_{i}={\begin{cases}-{\dfrac {1}{h^{2}}}\left\{[u]+h_{R}\left[u_{x}\right]+{\dfrac {h_{R}^{2}}{2}}\left[u_{xx}\right]+b\left(2h_{R}^{3}\left[u_{xxx}\right]+{\dfrac {h_{R}^{4}}{2}}\left[u_{xxxx}\right]\right)\right\},&i=I,\\[3mm]{\dfrac {1}{h^{2}}}\left\{[u]+h_{L}\left[u_{x}\right]+{\dfrac {h_{L}^{2}}{2}}\left[u_{xx}\right]+b\left(2h_{L}^{3}\left[u_{xxx}\right]+{\dfrac {h_{L}^{4}}{2}}\left[u_{xxxx}\right]\right)\right\},&i=I+1,\end{cases}}$

(7)

and where ${\textstyle b=1/12}$.

Proof.

We obtain a third-order scheme for at ${\textstyle x_{I}}$ and ${\textstyle x_{I+1}}$ following similar ideas as the ones developed for the generalized Taylor expansion proposed by Xu & Wang [36] and the IIM for elliptic interface problems with straight interfaces proposed by Feng & Li [37]. The idea is to consider extended smooth solutions of ${\textstyle u}$ such that we can apply the standard central scheme to ${\textstyle x_{I}}$ and ${\textstyle x_{I+1}}$. For instance, a function based on the original left solution is defined as

$u_{\ell }(x)={\begin{cases}u(x),&x\leq x_{\alpha },\\\displaystyle u^{-}+h_{R}u_{x}^{-}+{\frac {h_{R}^{2}}{2}}u_{xx}^{-}+{\dfrac {h_{R}^{3}}{6}}u_{xxx}^{-}+{\dfrac {h_{R}^{4}}{24}}u_{xxxx}^{-},&x_{\alpha }{<}x,\\\end{cases}}$

Using Taylor series expansions of ${\textstyle u_{I+1}}$ around ${\textstyle x_{\alpha }}$, the definition of jumps (5), and some simplification yield

$\left.\delta ^{2}u\right|_{I}=\left.\delta ^{2}u_{\ell }\right|_{I}+{\dfrac {1}{h^{2}}}\left([u]+h_{R}\left[u_{x}\right]+{\dfrac {h_{R}^{2}}{2}}\left[u_{xx}\right]+{\dfrac {h_{R}^{3}}{6}}\left[u_{xxx}\right]+{\dfrac {h_{R}^{4}}{24}}\left[u_{xxxx}\right]\right)+O(h^{3}).$

Thus,

$\left.\delta ^{2}u\right|_{I}=\left.(u_{\ell })_{xx}\right|_{I}+bh^{2}\left.(u_{\ell })_{xxxx}\right|_{I}$ $+{\dfrac {1}{h^{2}}}\left([u]+h_{R}\left[u_{x}\right]+{\dfrac {h_{R}^{2}}{2}}\left[u_{xx}\right]+{\dfrac {h_{R}^{3}}{6}}\left[u_{xxx}\right]+{\dfrac {h_{R}^{4}}{24}}\left[u_{xxxx}\right]\right)+O(h^{3}).$

Finally, we get (6) which complete the proof. The same procedure can be applied for the proof at ${\textstyle x_{I+1}}$. We refer to the reader to the work of Itza Balam and Uh Zapata [14] for more details. ♦

Remark 1: If ${\textstyle b=0}$, then Theorem 1 yields the standard second-order finite-difference method for regular points given by

$\left.u_{xx}\right|_{i}=\left.\delta ^{2}u\right|_{i}+O(h^{2}),\quad i\neq I,I+1,$

(8)

and Theorem 1 results in an IIM of first-order for irregular points ${\textstyle i=I,I+1}$ as follows

$\left.u_{xx}\right|_{i}=\left.\delta ^{2}u\right|_{i}+c_{i}+O(h),\quad i=I,I+1,$

(9)

where

$c_{i}={\begin{cases}-{\dfrac {1}{h^{2}}}\left([u]+h_{R}\left[u_{x}\right]+{\dfrac {h_{R}^{2}}{2}}\left[u_{xx}\right]\right),&i=I,\\[3mm]{\dfrac {1}{h^{2}}}\left([u]+h_{L}\left[u_{x}\right]+{\dfrac {h_{L}^{2}}{2}}\left[u_{xx}\right]\right),&i=I+1.\end{cases}}$

(10)

Note that, in this case, we only require to explicitly know jump conditions ${\textstyle [u]}$, ${\textstyle \left[u_{x}\right]}$ and ${\textstyle \left[u_{xx}\right]}$.

2.2 Implicit approximation for a real-valued function

Before applying previous results to approximate differential equations, it would be useful to express the operator ${\textstyle {\mathfrak {D}}^{2}u}$ using finite differences for a real-valued function ${\textstyle u}$ rather than its second-order derivative ${\textstyle u_{xx}}$. The finite-difference formula is obtained by approximating ${\textstyle \left.u_{xx}\right|}$ with the central finite differences, as presented in (1)-(10) for regular and irregular points, respectively.

For regular points (${\textstyle i\neq I,I+1}$), it follows from equation (1):

$\left.{\mathfrak {D}}^{2}u\right|_{i}=u_{i}+\left.bh^{2}u_{xx}\right|_{i}=u_{i}+bh^{2}\left\{\left.\delta ^{2}u\right|_{i}+O(h^{2})\right\}=bu_{i-1}+\,(1-2b)u_{i}+\,bu_{i+1}+O(h^{4}).$

Thus, if we define as ${\textstyle {\mathfrak {d}}^{2}}$ the resulting finite-difference

$\left.{\mathfrak {d}}^{2}u\right|_{i}:=bu_{i-1}+\,(1-2b)u_{i}+\,bu_{i+1},$

(11)

then, we have that

$\left.{\mathfrak {D}}^{2}u\right|_{i}=\left.{\mathfrak {d}}^{2}u\right|_{i}+O(h^{4}),\quad i\neq I,I+1.$

(12)

However, we still have the same issue to overcome for a discontinuous function ${\textstyle u}$. We remark that this second-order derivative of ${\textstyle u}$ is already multiplied for ${\textstyle h^{2}}$. Consequently, we can use the IIM technique where the contribution term is only first-order accurate to ${\textstyle u}$; thus, we still have a local ${\textstyle O(h^{3})}$ to keep a global fourth-order accurate method. Then for irregular points, the IIM applied for this term follows

$\left.{\mathfrak {D}}^{2}u\right|_{i}=\left.{\mathfrak {d}}^{2}u\right|_{i}+(c_{u})_{i}+O(h^{3}),\quad i=I,I+1,$

(13)

where ${\textstyle {\mathfrak {d}}^{2}}$ is given by finite difference (11) and

$(c_{u})_{i}={\begin{cases}-b\left([u]+h_{R}\left[u_{x}\right]+{\dfrac {h_{R}^{2}}{2}}\left[u_{xx}\right]\right),&i=I,\\[3mm]b\left([u]+h_{L}\left[u_{x}\right]+{\dfrac {h_{L}^{2}}{2}}\left[u_{xx}\right]\right),&i=I+1.\end{cases}}$

(14)

Finally, for regular and irregular points, we have high-order finite-difference approximations of the implicit operator ${\textstyle {\mathfrak {D}}^{2}}$ applied to a real-valued function ${\textstyle u}$ as in (12) and (13), and to its second-order derivative ${\textstyle u_{xx}}$ as in (2) and (6). Now, we proceed to implement them to the solution of differential equations, as presented in the following two sections.

3 Poisson equation

In this section, we develop a fourth-order finite difference scheme for the Poisson equation. Let us consider ${\textstyle u}$ and ${\textstyle f}$ as the solution of the problem and known right-hand side function, respectively. Thus the interface problem is given by

$\Delta u({\boldsymbol {x}})=f({\boldsymbol {x}}),\quad {\boldsymbol {x}}\in \Omega \setminus \Gamma ,\quad \Omega =\Omega ^{+}\cup \Omega ^{-},$ $u({\boldsymbol {x}})=g({\boldsymbol {x}}),\quad {\boldsymbol {x}}\in \partial \Omega ,$ $\left[u\right]_{\Gamma }={\mathfrak {p}}({\boldsymbol {x}}),\quad {\boldsymbol {x}}\in \Gamma ,$ $\left[u_{\boldsymbol {n}}\right]_{\Gamma }={\mathfrak {q}}({\boldsymbol {x}}),\quad {\boldsymbol {x}}\in \Gamma {.}$

We divide ${\textstyle \Omega }$ in two regions, ${\textstyle \Omega ^{+}}$ and ${\textstyle \Omega ^{-}}$, separated by an immersed interface ${\textstyle \Gamma }$. Dirichlet boundary conditions are defined on ${\textstyle \partial \Omega }$ through function ${\textstyle g}$. We assume that ${\textstyle u}$ and ${\textstyle f}$ may have discontinuities at the interface ${\textstyle \Gamma }$. Thus, we require additional conditions ${\textstyle \left[u\right]_{\Gamma }}$ and ${\textstyle \left[u_{\boldsymbol {n}}\right]_{\Gamma }}$ known as principal jump conditions. Here, ${\textstyle u_{\boldsymbol {n}}}$ is the derivative in the normal direction. Note that the principal jump conditions, ${\textstyle {\mathfrak {p}}}$ and ${\textstyle {\mathfrak {q}}}$, are known functions defined on ${\textstyle \Gamma }$.

In the context of the general problem, the computational domain can be considered into multiple dimensions. Nevertheless, since the primary objective of this paper is to illustrate the fundamental attributes of the proposed implicit high-order method, we concentrate on investigating the one-dimensional (1D) interface problem as defined by

$u_{xx}(x)=f(x),\quad x\in ({\mathfrak {a}},x_{\alpha })\cup (x_{\alpha },{\mathfrak {b}}),$ (15) $u(x)=g(x),\quad x\in \{{\mathfrak {a}},{\mathfrak {b}}\},$ (16) $\left[u\right]={\mathfrak {p}},$ (17) $\left[u_{x}\right]={\mathfrak {q}},$ (18)

Here, ${\textstyle u}$ and ${\textstyle f}$ can be discontinuous functions at a given fixed point ${\textstyle x=x_{\alpha }}$, and principal jump conditions ${\textstyle \left[u\right]=\left[u\right]_{x_{\alpha }}}$ and ${\textstyle \left[u_{x}\right]=\left[u_{x}\right]_{x_{\alpha }}}$ are known values at ${\textstyle x_{\alpha }}$.

3.1 IFD-IIM for the 1D Poisson problem

Let us consider that ${\textstyle x_{\alpha }}$ is located between the adjacent grid points ${\textstyle x_{I}}$ and ${\textstyle x_{I+1}}$ as ${\textstyle x_{I}\leq x_{\alpha }<x_{I+1}}$. If we apply operator ${\textstyle {\mathfrak {D}}^{2}(\cdot )=(\cdot )+bh^{2}(\cdot )_{xx}}$ at both sides of (15), then we get

$\left.{\mathfrak {D}}^{2}u_{xx}\right|_{i}=\left.{\mathfrak {D}}^{2}f\right|_{i},\qquad i=2,\dots ,N.$

(19)

For ${\textstyle i=1}$ and ${\textstyle i=N+1}$, the grid points are at the boundary, thus the Dirichlet boundary condition can be directly applied. Thus, using the IFD scheme (2) and approximation (12) in (19) at ${\textstyle x_{i}}$, we have that the finite-difference scheme for regular points is given by

$\left.\delta ^{2}u\right|_{i}=\left.{\mathfrak {d}}^{2}f\right|_{i}+O(h^{4}),\qquad i\neq I,I+1.$

(20)

Similarly, using formulas (6) and (13), the scheme for irregular points is given by

$\left.\delta ^{2}u\right|_{i}+C_{i}=\left.{\mathfrak {d}}^{2}f\right|_{i}+(c_{f})_{i}+O(h^{3}),\qquad i=I,I+1,$

(21)

where ${\textstyle C_{i}}$ and ${\textstyle (c_{f})_{i}}$ are given by (7) and (14), respectively. Thus, combining the results for all grid points, the IFD-IIM for the 1D Poisson equation (15) at ${\textstyle x_{i}}$ is given by

${\frac {1}{h^{2}}}U_{i-1}\,\,-{\frac {2}{h^{2}}}U_{i}\,+\,{\frac {1}{h^{2}}}U_{i+1}=bf_{i-1}+\,(1-2b)f_{i}+\,bf_{i+1}+{\mathfrak {C}}_{i},\quad i=2,\dots ,N,$

(22)

where total contribution is ${\textstyle {\mathfrak {C}}_{i}=(c_{f})_{i}-C_{i}}$ with

$C_{i}={\begin{cases}-{\dfrac {1}{h^{2}}}\left([u]+h_{R}\left[u_{x}\right]+{\dfrac {h_{R}^{2}}{2}}\left[u_{xx}\right]+b\left(2h_{R}^{3}\left[u_{xxx}\right]+{\dfrac {h_{R}^{4}}{2}}\left[u_{xxxx}\right]\right)\right),&i=I,\\[4mm]{\dfrac {1}{h^{2}}}\left([u]+h_{L}\left[u_{x}\right]+{\dfrac {h_{L}^{2}}{2}}\left[u_{xx}\right]+b\left(2h_{L}^{3}\left[u_{xxx}\right]+{\dfrac {h_{L}^{4}}{2}}\left[u_{xxxx}\right]\right)\right),&i=I+1,\\[2mm]0,&{\textrm {elsewhere,}}\end{cases}}$

(23)

and

$(c_{f})_{i}={\begin{cases}-b\left([f]+h_{R}\left[f_{x}\right]+{\dfrac {h_{R}^{2}}{2}}\left[f_{xx}\right]\right),&i=I,\\[3mm]b\left([f]+h_{L}\left[f_{x}\right]+{\dfrac {h_{L}^{2}}{2}}\left[f_{xx}\right]\right),&i=I+1,\\[2mm]0,&{\textrm {elsewhere.}}\end{cases}}$

(24)

We remark that scheme (22)-(24) results in an approximation with local truncation error of fourth- and third-order for regular and irregular grid points, respectively. Thus, a global ${\textstyle O(h^{4})}$ method is expected.

3.2 IFD-IIM and principal jump conditions

Note that ${\textstyle \left[u\right]}$, ${\textstyle \left[u_{x}\right]}$, ${\textstyle \left[u_{xx}\right]}$, ${\textstyle \left[u_{xxx}\right]}$, and ${\textstyle \left[u_{xxxx}\right]}$ must be known to apply the proposal fourth-order IFD-IIM. Thus, it seems that more jump conditions of ${\textstyle u}$ rather than the principal jump conditions (17) and (18) are required to have a fourth-order accurate method. However, we can use the Poisson equation (15) to obtain them as follows

$\left[u_{xx}\right]=[f],\qquad \left[u_{xxx}\right]=\left[f_{x}\right],\qquad \left[u_{xxxx}\right]=\left[f_{xx}\right].$

(25)

Thus, the total jump contribution for the one-dimensional Poisson problem is given by

${\mathfrak {C}}_{i}={\begin{cases}{\dfrac {1}{h^{2}}}\left([u]+h_{R}\left[u_{x}\right]+{\dfrac {h_{R}^{2}}{2}}\left[f\right]\right)&\\\qquad -b\left\{[f]+h_{R}\left(-{\dfrac {2h_{R}^{2}}{h^{2}}}+1\right)\left[f_{x}\right]+{\dfrac {h_{R}^{2}}{2}}\left(-{\dfrac {h_{R}^{2}}{h^{2}}}+1\right)\left[f_{xx}\right]\right\},&i=I,\\[2mm]-{\dfrac {1}{h^{2}}}\left([u]+h_{L}\left[u_{x}\right]+{\dfrac {h_{L}^{2}}{2}}\left[f\right]\right)&\\\qquad +b\left\{[f]+h_{L}\left(-{\dfrac {2h_{L}^{2}}{h^{2}}}+1\right)\left[f_{x}\right]+{\dfrac {h_{L}^{2}}{2}}\left(-{\dfrac {h_{L}^{2}}{h^{2}}}+1\right)\left[f_{xx}\right]\right\},&i=I+1,\\[4mm]0,&{\textrm {elsewhere}}.\end{cases}}$

(26)

Thus contribution ${\textstyle {\mathfrak {C}}_{i}}$ depends only on the principal jump conditions, ${\textstyle [u]}$ and ${\textstyle [u_{x}]}$, and right-hand side jumps: ${\textstyle [f]}$, ${\textstyle \left[f_{x}\right]}$, and ${\textstyle \left[f_{xx}\right]}$. The additional jumps derivatives from the right-hand side can be approximated using the current values of ${\textstyle f}$. In this paper, we will assume that we know them.

Remark 2: For the 1D Poisson problem, a global second-order IIM (${\textstyle b=0}$) only requires to know the principal jump conditions ${\textstyle [u]}$, ${\textstyle \left[u_{x}\right]}$ and ${\textstyle \left[f\right]}$.

Remark 3: If ${\textstyle h_{R}/h=1}$, then ${\textstyle h_{L}=0}$ and both weight terms next to second-order derivative jump of ${\textstyle f}$ are equal to zero in (26). Thus, we do not require to know jump condition ${\textstyle \left[f_{xx}\right]}$ to obtain a fourth-order method when the interface is located at a grid point.

4 Heat equation with a fixed interface

For the second differential equation, we consider the heat conduction equation with initial, boundary, and principal jump conditions, as follows

$u_{t}({\boldsymbol {x}},t)=\Delta u({\boldsymbol {x}},t)-f({\boldsymbol {x}},t),\quad {\boldsymbol {x}}\in \Omega \setminus \Gamma \times (0,T],\quad \Omega =\Omega ^{+}\cup \Omega ^{-},$ $u({\boldsymbol {x}},0)=u_{0}({\boldsymbol {x}}),\quad {\boldsymbol {x}}\in \Omega ,$ $u({\boldsymbol {x}},t)=g({\boldsymbol {x}},t),\quad ({\boldsymbol {x}},t)\in \partial \Omega \times [0,T],$ $\left[u\right]_{\Gamma }={\mathfrak {p}}({\boldsymbol {x}},t),\quad {\boldsymbol {x}}\in \Gamma \times [0,T],$ $\left[u_{\boldsymbol {n}}\right]_{\Gamma }={\mathfrak {q}}({\boldsymbol {x}},t),\quad {\boldsymbol {x}}\in \Gamma \times [0,T],$

where the source ${\textstyle f}$ and initial value ${\textstyle u_{0}}$ may be discontinuous or singular across a fixed interface ${\textstyle \Gamma }$. The interface ${\textstyle \Gamma }$ is immersed in the domain ${\textstyle \Omega }$ and divides into two parts, ${\textstyle \Omega ^{+}}$ and ${\textstyle \Omega ^{-}}$. As the Poisson equation, ${\textstyle g}$, ${\textstyle {\mathfrak {p}}}$ and ${\textstyle {\mathfrak {q}}}$ are known functions. This paper only focuses on the one-dimensional problem given by

$u_{t}=u_{xx}-f,\quad (x,t)\in ({\mathfrak {a}},x_{\alpha })\cup (x_{\alpha },{\mathfrak {b}})\times (0,T],$ (27) $u(x,0)=u_{0}(x),\quad x\in ({\mathfrak {a}},{\mathfrak {b}}),$ (28) $u(x,t)=g(x,t),\quad (x,t)\in \{{\mathfrak {a}},{\mathfrak {b}}\}\times [0,T],$ (29) $\left[u\right]={\mathfrak {p}}(t),\quad t\in [0,T],$ (30) $\left[u_{x}\right]={\mathfrak {q}}(t),\quad t\in [0,T].$ (31)

where the source ${\textstyle f}$ and initial value ${\textstyle u_{0}}$ may be discontinuous or singular across a fixed interface located at ${\textstyle x_{\alpha }}$.

4.1 IFD-IIM for the 1D heat conduction problem

Since the interface is fixed and all the quantities are continuous with time, we can approximate the time derivative using the Crank-Nicolson method, as follows

$u^{n+1}=u^{n}+{\dfrac {\Delta t}{2}}\left\{\left(u_{xx}-f\right)^{n+1}+\left(u_{xx}-f\right)^{n}\right\}+O(\Delta t^{2}).$

(32)

Applying the fourth-order operator (3) to equation (32) for ${\textstyle i=2,\dots ,N}$, yields

$\left.{\mathfrak {D}}^{2}u\right|_{i}^{n+1}=\left.{\mathfrak {D}}^{2}u\right|_{i}^{n}+{\dfrac {\Delta t}{2}}\left\{\left.{\mathfrak {D}}^{2}u_{xx}\right|_{i}^{n+1}+\left.{\mathfrak {D}}^{2}u_{xx}\right|_{i}^{n}-\left.{\mathfrak {D}}^{2}f\right|_{i}^{n+1}-\left.{\mathfrak {D}}^{2}f\right|_{i}^{n}\right\}+O(\Delta t^{2}).$

(33)

By employing the IFD method (2) and approximation (12), equation (33) can be approximated for regular points as follows

$\left.{\mathfrak {d}}^{2}u\right|_{i}^{n+1}=\left.{\mathfrak {d}}^{2}u\right|_{i}^{n}+{\dfrac {\Delta t}{2}}\left\{\left.\delta ^{2}u\right|_{i}^{n+1}+\left.\delta ^{2}u\right|_{i}^{n}-\left.{\mathfrak {d}}^{2}f\right|_{i}^{n+1}-\left.{\mathfrak {d}}^{2}f\right|_{i}^{n}\right\}$ $+O(\Delta t^{2})+O(h^{4}),\qquad i\neq I,I+1.$

(34)

By using the IFD-IIM (6) and approximation (13), the implicit scheme for irregular points is given by

$\left.{\mathfrak {d}}^{2}u\right|_{i}^{n+1}=\left.{\mathfrak {d}}^{2}u\right|_{i}^{n}+{\dfrac {\Delta t}{2}}\left\{\left.\delta ^{2}u\right|_{i}^{n+1}+\left.\delta ^{2}u\right|_{i}^{n}-\left.{\mathfrak {d}}^{2}f\right|_{i}^{n+1}-\left.{\mathfrak {d}}^{2}f\right|_{i}^{n}\right\}+{\mathfrak {C}}_{i}$ $+O(\Delta t^{2})+O(h^{3}),\qquad i=I,I+1,$

(35)

where

${\mathfrak {C}}_{i}=-(c_{u})_{i}^{n+1}+(c_{u})_{i}^{n}+{\dfrac {\Delta t}{2}}\left\{C_{i}^{n+1}+C_{i}^{n}-(c_{f})_{i}^{n+1}-(c_{f})_{i}^{n}\right\}.$

(36)

Here, ${\textstyle (c_{u})_{i}}$ and ${\textstyle (c_{f})_{i}}$ are defined as (14), and ${\textstyle C_{i}}$ is given by (7). Thus, for 1D heat conduction equation (27), the IFD-IIM reads

$(b-r)U_{i-1}^{n+1}+(1-2b+2r)U_{i}^{n+1}+(b-r)U_{i+1}^{n+1}$ $=(b+r)U_{i-1}^{n}+(1-2b-2r)U_{i}^{n}+(b+r)U_{i+1}^{n}$ $-{\dfrac {\Delta t}{2}}\left\{bf_{i-1}^{n+1}+(1-2b)f_{i}^{n+1}+bf_{i+1}^{n+1}\right\}$ (37) $-{\dfrac {\Delta t}{2}}\left\{bf_{i-1}^{n}+(1-2b)f_{i}^{n}+bf_{i+1}^{n}\right\}+{\mathfrak {C}}_{i},$

where ${\textstyle r={\frac {1}{2}}\Delta t/h^{2}}$ and ${\textstyle {\mathfrak {C}}_{i}}$ is given by (36). Here, we have that ${\textstyle {\mathfrak {C}}_{i}=0}$ for regular points.

Remark 4: The IFD-IIM (37) is unconditionally stable and the local truncation error is of ${\textstyle O(\Delta t^{2}+h^{4})}$ and ${\textstyle O(\Delta t^{2}+h^{3})}$ for regular and irregular grid points, respectively. Thus a global fourth-order method is expected by taking ${\textstyle \Delta t=O(h^{2})}$.

Remark 5: As the Crank-Nicolson method is implicit in time, the IFD-IIM solves a linear system at each time step. To address this efficiently, we employed Thomas' algorithm, given that the resulting matrix is tridiagonal. Furthermore, the implicit method in space preserves the original structure of this system of equations, thus yielding a higher-order method without compromising the efficiency of the standard scheme.

Remark 6: In addition to accounting for the contributions of the source term ${\textstyle f}$ and its derivatives, the finite-difference scheme presented in equation (37) requires additional knowledge of the jump conditions for the solution given by ${\textstyle \left[u_{xx}\right]}$, ${\textstyle \left[u_{xxx}\right]}$, and ${\textstyle \left[u_{xxxx}\right]}$. It is worth noting that, although not presented here, there are techniques available for deriving all the necessary jump conditions from the principal jump conditions [14].

5 Numerical results for the Poisson equation

In this section, we test the IFD-IIM for different examples of the Poisson equation. In the following simulations, we numerically solve the equation for a given right-hand side function and compare it with its analytic solution. In all cases the resulting linear system is solved using the Thomas' algorithm.

The numerical method is tested using three different examples. Example 1 considers a smooth solution to verify the fourth-order implicit method for smooth solutions. Example 2 studies a Poisson equation with a discontinuous solution in a single interface point. Finally, Example 3 presents a discontinuous problem with multiple interface points. A Matlab code of these examples can be download at https://github.com/CIMATMerida/IFD-IIM

For the all these examples, the computational domain is the interval ${\textstyle [0,1]}$, and the grid spacing is ${\textstyle h=1/N}$ for different ${\textstyle N}$ sub-intervals. The errors are reported using the ${\textstyle L_{\infty }}$-norm and the discrete ${\textstyle L_{2}}$-norm calculated as

$\left\|e\right\|_{\infty }=\max _{1\leq i\leq N+1}\left|u_{i}-U_{i}\right|,\quad \mathrm {and} \quad \left\|e\right\|_{2}=\left(\sum _{i=1}^{N+1}(u_{i}-U_{i})^{2}\Delta x\right)^{1/2},$

(38)

respectively, where ${\textstyle U_{i}}$ and ${\textstyle u_{i}}$ corresponds to the numerical and exact solution at ${\textstyle x_{i}}$, respectively. The estimated order of accuracy is computed as

${\textrm {Order}}:={\dfrac {\log \left(\left\|e\right\|_{N_{1}}/\left\|e\right\|_{N_{2}}\right)}{\log \left(N_{1}/N_{2}\right)}},$

(39)

where ${\textstyle N_{1}}$ and ${\textstyle N_{2}}$ indicates the different number of sub-intervals of the corresponding norm.

5.1 Example 1. Poison equation with smooth solution

Example 1 considers the one-dimensional Poisson problem

${\begin{array}{rcl}u_{xx}(x)&=&f(x),\quad x\in (0,1),\\[1mm]u(0)&=&e^{-\pi /4},\\u(1)&=&e^{-9\pi /4},\end{array}}$

(40)

where the right-hand side in (40) is given by

$f(x)=4\pi (16\pi x^{2}-8\pi x+\pi {-2)}e^{-4\pi (x-1/4)^{2}}.$

The exact solution of problem (40) is an infinitely smooth and differentiable function given by the following expression

$u(x)=e^{-4\pi (x-1/4)^{2}}.$

(41)

Note that the function ${\textstyle f}$ and Dirichlet boundary conditions are obtained directly from (41). Due to the regularity of the solution, the jump contributions ${\textstyle {\mathcal {C}}_{I}}$ and ${\textstyle {\mathcal {C}}_{I+1}}$ are equal to zero.

Table 1 presents the convergence analysis of Example 1 for different grid resolutions. If ${\textstyle b=0}$, then we recover the standard central finite-difference method of second-order accuracy. On the other hand, the fourth-order implicit scheme is recovered when ${\textstyle b=1/12}$. Last row of table 1 shows the numerical order calculated by the regression-line slope based on a least squares method (LSM) of the ${\textstyle L_{\infty }}$- and ${\textstyle L_{2}}$-norm error. A complete analysis of the IFD method for smooth solutions can be found in [12].

Table. 1 Convergence analysis of Example 1 testing a Poisson equation with smooth solution.

	${\textstyle b=0}$				${\textstyle b=1/12}$
${\textstyle N}$	${\textstyle L_{\infty }}$-norm	Order	${\textstyle L_{2}}$-norm	Order	${\textstyle L_{\infty }}$-norm	Order	${\textstyle L_{2}}$-norm	Order
10	7.52e-02	–-	2.84e-02	–-	9.30e-03	–-	3.56e-03	–-
20	1.70e-02	2.15	6.52e-03	2.12	5.05e-04	4.20	1.93e-04	4.21
40	4.15e-03	2.03	1.60e-03	2.03	3.06e-05	4.04	1.17e-05	4.04
80	1.03e-03	2.01	3.98e-04	2.01	1.90e-06	4.01	7.27e-07	4.01
160	2.57e-04	2.00	9.95e-05	2.00	1.18e-07	4.00	4.54e-08	4.00
LSM		2.04		2.04		4.06		4.06

5.2 Example 2. Poison equation with a single interface

For Example 2, we show the method's capability by solving a single interface problem located at ${\textstyle x=x_{\alpha }}$. Here, we consider the one-dimensional Poisson problem

${\begin{array}{rcl}u_{xx}(x)&=&f(x),\quad x\in (0,x_{\alpha })\cup (x_{\alpha },1),\\[1mm]u(0)&=&0\\[1mm]u(1)&=&-1,\\[1mm]\left[u\right]&=&\cos(\pi x_{\alpha })-\sin(\pi x_{\alpha }),\\[1mm]\left[u_{x}\right]&=&-\pi (\sin(\pi x_{\alpha })+\cos(\pi x_{\alpha })),\end{array}}$

(42)

where the right-hand side in (42) is

$f(x)={\begin{cases}-\pi ^{2}\sin(\pi x),&x<x_{\alpha },\\-\pi ^{2}\cos(\pi x),&x>x_{\alpha }{.}\end{cases}}$

The exact solution of problem (42) is a discontinuous function given by the expression

$u(x)={\begin{cases}\sin(\pi x),&x<x_{\alpha },\\\cos(\pi x),&x>x_{\alpha }{.}\end{cases}}$

(43)

Note that the right-hand side, Dirichlet boundary conditions, and principal jump conditions are obtained directly from equation (43).

We test two different points: ${\textstyle x_{\alpha }=0.4}$ and ${\textstyle x_{\alpha }=0.63}$. For the case ${\textstyle x_{\alpha }=0.4}$, we always have ${\textstyle h_{R}/h=1}$ for ${\textstyle N=10\times {2}^{n}}$ (${\textstyle n=0,1,2,\dots }$); thus the interface is always located at one grid point of that resolution. In general, for ${\textstyle x_{\alpha }=0.63}$, we have different ${\textstyle h_{R}/h}$ values for different ${\textstyle N}$ numbers. Fig. 2 shows the numerical and exact solution when the interface is located at these two values using ${\textstyle N=40}$. As expected, the exact solution is accurately recovered for both cases.

Figure 2: Numerical and exact solution of Example 2 using $N=40$ using (a) $x_{\alpha }=0.4$, and (b) $x_{\alpha }=0.63$. Here, the interface is located at one grid point using $N=40$ and $x_{\alpha }=0.4$.

Table 2 shows the convergence analysis for Example 2 for the two ${\textstyle x_{\alpha }}$ values. Observe that a second-order method is recovered for ${\textstyle b=0}$ and a fourth-order method for ${\textstyle b=1/12}$. As expected, the IFD-IIM numerical order does not depend on the location of the interface. However, the magnitude of the error may present minor variations due to the interface position. Fig. 3 shows the error analysis corresponding to interface locations ${\textstyle x_{\alpha }=0.40}$ and ${\textstyle x_{\alpha }=0.63}$ for ${\textstyle N=10,11,12,\dots ,320}$. The solid black line for this and the following figures shows the numerical order calculated by the regression-line slope based on a least squares method of the ${\textstyle L_{\infty }}$-norm.

Table. 2 Convergence analysis of Example 2 using the IFD-IIM.

	${\textstyle b=0}$				${\textstyle b=1/12}$
${\textstyle N}$	${\textstyle L_{\infty }}$-norm	Order	${\textstyle L_{2}}$-norm	Order	${\textstyle L_{\infty }}$-norm	Order	${\textstyle L_{2}}$-norm	Order
10	1.69e-02	–-	5.19e-03	–-	5.75e-05	–-	3.42e-05	–-
20	4.21e-03	2.00	1.18e-03	2.13	3.58e-06	4.00	1.97e-06	4.12
40	1.05e-03	2.00	3.07e-04	1.95	2.24e-07	4.00	1.39e-07	3.82
80	2.63e-04	2.00	7.53e-05	2.03	1.40e-08	4.00	7.75e-09	4.16
160	6.57e-05	2.00	1.86e-05	2.01	8.74e-10	4.00	5.37e-10	3.85
LSM		2.04		2.02		4.00		3.99

Figure 3: Convergence analysis of Example 2 for $N=10,\dots ,320$ using (a) $x_{\alpha }=0.40$, and (b) $x_{\alpha }=0.63$. The solid black line shows the numerical order $4.00$ calculated by the regression-line slope based on a least squares method of the $L_{\infty }$-norm.

The contribution formula includes jumps ${\textstyle [u]}$, ${\textstyle [u_{x}]}$, ${\textstyle [u_{xx}]}$, ${\textstyle [u_{xxx}]}$, and ${\textstyle [u_{xxxx}]}$ to obtain a fourth-order accurate method. Fig. 4 shows that if we add additional jumps of high-order derivatives into ${\textstyle {\mathcal {C}}}$, such as ${\textstyle [u_{xxxxx}]=[f_{xxx}]}$, we observe that the error oscillation decreases in comparison with Fig. 3 results. It is expected because the method is ${\textstyle O(h^{4})}$ for the whole computational domain, including the irregular points. Thus, we can mitigate error oscillations due to interface position by adding high-order jumps.

Figure 4: Convergence analysis of Example 2 for $N=10,\dots ,320$ using (a) $\alpha =0.40$, and (b) $\alpha =0.63$. The contribution includes jumps up to fifth-order ($[u_{xxxxx}]=[f_{xxx}]$).

Now we study the effect of removing jumps in ${\textstyle {\mathcal {C}}}$. Let us consider a contribution term up to the third derivative jump by dropping ${\textstyle [u_{xxxx}]=[f_{xx}]}$. Thus, the numerical approximation is only third-order accurate, as shown in Fig. 5 for different interface values. Note that error oscillations may have a clear pattern or a scattered distribution depending on the interface location. In general, the error magnitude perturbations are related to the variations coming from ${\textstyle h_{R}/h}$ and ${\textstyle h_{L}/h}$ values in the ${\textstyle {\mathcal {C}}}$ formula (26).

Figure 5: Convergence analysis of Example 2 for $N=10,\dots ,320$ using (a) $\alpha =0.40$, and (b) $\alpha =0.63$. The contribution includes jumps up to third-order ($[u_{xxx}]=[f_{x}]$).

Note that there are some points in Fig. 5, marked with circles, that are close to the fourth-order line. Those circled markers correspond to the values with ${\textstyle h_{R}/h=1}$: (a) ${\textstyle N=5,10,15,\dots ,315}$ for ${\textstyle x_{\alpha }=0.40}$, and (b) ${\textstyle N=100,200,300}$ for ${\textstyle x_{\alpha }=0.63}$. Both set of points satisfy that the interface ${\textstyle x_{\alpha }}$ is located at a grid point ${\textstyle x_{I}}$. In the case of ${\textstyle x_{\alpha }=0.4}$, we observe that the global order (black points) is close to ${\textstyle 3.21}$; meanwhile, the circled points order is four. Similar behavior is obtained for ${\textstyle x_{\alpha }=0.63}$. Thus, for a given mesh resolution ${\textstyle N}$, the IFD-IIM is still fourth-order accurate for ${\textstyle h_{R}/h=1}$, as proven theoretically in Section 4.2.

5.3 Example 3. Poison equation with multiple interface points

Example 3 investigates the numerical scheme capability to solve a multiple interface problem. We only focus on two interface points located at ${\textstyle x=x_{\alpha _{1}}}$ and ${\textstyle x=x_{\alpha _{2}}}$. However, the methodology could be applied for multiple interfaces by doing minor modifications in the implementation. For this problem, the exact solution of the Poisson problem is given by

$u(x)={\begin{cases}\sin(2\pi x),&x<x_{\alpha _{1}},\\\cos(2\pi x),&x_{\alpha _{1}}<x<x_{\alpha _{2}},\\\sin(2\pi x),&x_{\alpha _{2}}<x.\end{cases}}$

(44)

The right-hand function, Dirichlet boundary conditions, and principal jump conditions are obtained directly from the exact solution (44). We consider the same computational domain and grid resolution as previous 1D examples. Fig. 6 presents the analytical and numerical solution when the interface is located at ${\textstyle x_{\alpha _{1}}=0.3}$ and ${\textstyle x_{\alpha _{2}}=0.7}$ using ${\textstyle N=40}$. This figure also shows the error analysis. As expected, the IFD-IIM is a fourth-order accurate method.

Figure 6: (a) Numerical and exact solution of Example 3 with multiple interfaces using $N=40$, and (b) convergence error analysis using grid resolutions of $N=10$, 20, 40, 80 and 160.

6 Numerical results for the Heat equation

This section tests the IFD-IIM for the Heat conduction equation in different scenarios. Example 4 verifies the fourth-order implicit method for smooth solutions in space. Example 5 studies a Heat equation with a discontinuous solution in a single interface point and no source term. Finally, Example 6 presents a general discontinuous problem. For the all these examples, the computational domain is the same interval ${\textstyle [0,1]}$ and space step, ${\textstyle h}$, used for the Poisson examples. Here, the final time is at ${\textstyle T=0.5}$ and the time step is ${\textstyle \Delta t={\frac {1}{4}}h^{2}}$. The error and estimated order of accuracy are reported using (38) and (39), respectively, at the final step approximation.

6.1 Example 4. Heat equation with smooth solution

This example considers the heat conduction problem

${\begin{array}{rcll}u_{t}(x,t)&=&u_{xx}(x,t)-f(x,t),&(x,t)\in (0,1)\times (0,{\frac {1}{2}}],\\[2mm]u(x,0)&=&\sin \left({\frac {3}{2}}\pi x\right),&x\in (0,1),\\[2mm]u(0,t)&=&0,&t\in [0,{\frac {1}{2}}],\\[2mm]u(1,t)&=&-e^{-t},&t\in [0,{\frac {1}{2}}],\end{array}}$

(45)

where ${\textstyle f}$ in (45) is given by

$f(x,t)=\left(1-{\frac {9}{4}}\pi ^{2}\right)\sin \left({\frac {3}{2}}\pi x\right)e^{-t}.$

This example is constructed so that the exact solution is

$u(x,t)=\sin \left({\frac {3}{2}}\pi x\right)e^{-t}.$

(46)

Note that the source term, ${\textstyle f}$, initial condition and boundary conditions are obtained from the exact solution (46). The convergence analysis of this example is presented in Table 3 with the final time being ${\textstyle T=0.5}$. As expected, the high-order methods reach their corresponding order of accuracy.

Table. 3 Convergence analysis of Example 4 testing a heat equation with smooth solution.

	${\textstyle b=0}$				${\textstyle b=1/12}$
${\textstyle N}$	${\textstyle L_{\infty }}$-norm	Order	${\textstyle L_{2}}$-norm	Order	${\textstyle L_{\infty }}$-norm	Order	${\textstyle L_{2}}$-norm	Order
10	1.66e-02	–-	1.07e-02	–-	1.84e-04	–-	1.18e-04	–-
20	4.12e-03	2.01	2.65e-03	2.01	1.14e-05	4.01	7.34e-06	4.01
40	1.03e-03	2.00	6.60e-04	2.00	7.15e-07	4.00	4.58e-07	4.00
80	2.58e-04	2.00	1.65e-04	2.00	4.47e-08	4.00	2.86e-08	4.00
160	6.44e-05	2.00	4.12e-05	2.00	2.79e-09	4.00	1.79e-09	4.00
LSM		2.00		2.00		4.00		4.00

6.2 Example 5. Heat equation with discontinuous solution

In this example, the heat equation with discontinuous solution is given by

${\begin{array}{rcll}u_{t}(x,t)&=&u_{xx}(x,t),&(x,t)\in (0,x_{\alpha })\cup (x_{\alpha },1)\times (0,{\frac {1}{2}}],\\[2mm]u(x,0)&=&{\begin{cases}\sin(x),&x<x_{\alpha },\\\cos(x),&x>x_{\alpha },\\\end{cases}}&x\in (0,x_{\alpha })\cup (x_{\alpha },1),\\\\[-3mm]u(0,t)&=&0,&t\in [0,{\frac {1}{2}}],\\[2mm]u(1,t)&=&\cos(1)e^{-t},&t\in [0,{\frac {1}{2}}],\\[2mm]\left[u\right](t)&=&\left(\cos(x_{\alpha })-\sin(x_{\alpha })\right)e^{-t},&t\in [0,{\frac {1}{2}}],\\[2mm]\left[u_{x}\right](t)&=&-\left(\sin(x_{\alpha })+\cos(x_{\alpha })\right)e^{-t},&t\in [0,{\frac {1}{2}}].\end{array}}$

(47)

The exact solution ${\textstyle u}$ is defined as

$u(x,t)={\begin{cases}\sin(x)e^{-t},&x<x_{\alpha },\\\cos(x)e^{-t},&x>x_{\alpha }{.}\end{cases}}$

(48)

Similar to previous examples, we use ${\textstyle x_{\alpha }=0.4}$ and ${\textstyle x_{\alpha }=0.63}$. Note that the source term ${\textstyle f}$ of this problem is ${\textstyle f\equiv 0}$. In this example, both the function ${\textstyle u}$ and their derivatives have jumps, and these jumps vary in time. The boundary condition, initial condition, and all jumps are specified by ${\textstyle u}$.

The figure of numerical solution and absolute error plot using ${\textstyle N=40}$ for different time stages are shown in Fig. 7. On the other hand, the one-dimensional results at ${\textstyle T=0.5}$ are presented in Figs. 8 and 9. As expected, we can accurately recover the exact solution using the proposed method for ${\textstyle N=40}$ and considering different interface values. Additionally, we provide the absolute error for both cases, noting that the maximum error is found in close proximity to the point ${\textstyle x_{\alpha }}$. This behavior is also expected since the local truncation error for irregular points is less accurate than for regular points. Finally, the convergence analysis using ${\textstyle N=10}$, 20, 40, 80 and 160 confirms the fourth-order global convergence of the proposed method.

More details of the grid refinement analysis at ${\textstyle T=0.5}$ is presented in Table 4 for two different interface point locations. As expected, a fourth-order method is obtained for both interfaces. We remark that the variation of the errors using ${\textstyle x_{\alpha }=0.63}$ is because the different ${\textstyle h_{R}/h}$ values resulting from the discretization.

Figure 7: Numerical solution and absolute error of Example 5 using $N=40$, $x_{\alpha }=0.4$ for $t\in [0,0.5]$.

Figure 8: Numerical results at final time simulation $T=0.5$ and $x_{\alpha }=0.4$ of Example 5.

Figure 9: Numerical results at final time simulation $T=0.5$ and $x_{\alpha }=0.63$ of Example 5.

Table. 4 Convergence analysis of Example 5 using the IFD-IIM for the heat equation with a discontinuous solution.

	${\textstyle x_{\alpha }=0.4}$				${\textstyle x_{\alpha }=0.63}$
${\textstyle N}$	${\textstyle L_{\infty }}$-norm	Order	${\textstyle L_{2}}$-norm	Order	${\textstyle L_{\infty }}$-norm	Order	${\textstyle L_{2}}$-norm	Order
10	1.11e-07	–-	6.52e-08	–-	7.57e-08	–-	4.49e-08	–-
20	6.90e-09	4.01	4.01e-09	4.02	3.75e-09	4.34	2.25e-09	4.32
40	4.29e-10	4.00	2.49e-10	4.01	3.58e-10	3.39	2.09e-10	4.43
80	2.68e-11	4.00	1.55e-11	4.00	1.53e-11	4.55	8.93e-12	4.55
160	1.63e-12	4.03	9.39e-13	4.04	1.35e-12	3.50	7.82e-13	3.51
LSM		4.01		4.02		3.95		3.96

From Table 5, we can find the results when the time step is chosen as ${\textstyle \Delta t=h}$. As expected, the proposed IFD-IIM is stable and has two-order convergence either we take a second- or fourth-order approximation in space. We remark that this is a property of the selected time integration method. For instance, similar findings as Table 5 can be obtained for smooth solutions.

Table. 5 Convergence analysis of Example 5 testing the heat equation with $\Delta t=h$.

	${\textstyle b=0}$				${\textstyle b=1/12}$
${\textstyle N}$	${\textstyle L_{\infty }}$-norm	Order	${\textstyle L_{2}}$-norm	Order	${\textstyle L_{\infty }}$-norm	Order	${\textstyle L_{2}}$-norm	Order
10	2.94e-04	–-	1.87e-04	–-	3.78e-05	–-	2.67e-05	–-
20	8.53e-05	1.78	4.85e-05	1.95	1.03e-05	1.87	7.22e-06	1.89
40	2.11e-05	2.01	1.26e-05	1.94	2.74e-06	1.91	1.92e-06	1.91
80	5.31e-06	1.99	3.15e-06	2.00	6.91e-07	1.99	4.83e-07	1.99
160	1.34e-06	1.99	7.83e-07	2.00	1.72e-07	2.00	1.21e-07	2.00
LSM		1.96		1.97		1.95		1.95

6.3 Example 6. Heat equation with a general discontinuous solution

Finally, Example 6 investigates the capability to solve a heat interface problem with a general discontinuous solution. For this problem, we slightly modified the previous example such that the exact solution of the heat conduction problem is given by

$u(x)={\begin{cases}\sin(2\pi x)e^{-t},&x<x_{\alpha },\\\cos(2\pi x)e^{-t},&x>x_{\alpha }.\end{cases}}$

(49)

Here, the source term ${\textstyle f}$ is a general function that varies both in time and space. It is directly derived from equation (27) as well as the exact solution (49). Furthermore, the exact solution is utilized to specify the boundary condition, initial condition, and all jumps contributions.

In Fig. 10, we depict the temporal evolution of the numerical solution and absolute errors using parameters ${\textstyle N=40}$ and ${\textstyle x_{\alpha }=0.4}$. It is noteworthy that the behavior of the solution differs from the previous example. More in-depth analysis of the results at ${\textstyle T=0.5}$ are illustrated in Figs. 11 and 12, corresponding to values of ${\textstyle x_{\alpha }=0.4}$ and ${\textstyle x_{\alpha }=0.63}$, respectively. First, the numerical solution is accurately approximated using the proposed implicit method for ${\textstyle N=40}$, regardless of the interface locations. It is worth noting that the maximum error for ${\textstyle x_{\alpha }=0.63}$ is concentrated near to this point; however, this is not observed for ${\textstyle x_{\alpha }=0.4}$. This difference in behavior can be attributed to the interplay between the interface location (${\textstyle h_{R}/h=1}$ and ${\textstyle x_{\alpha }=0.4}$) and the non-zero right-hand side values, which contribute to the local truncation errors. Grid refinement analyses are also provided in these figures using ${\textstyle N=10}$, 20, 40, 80 and 160. As anticipated, the IFD-IIM method demonstrates fourth-order accuracy, a validation that is further detailed in Table 6.

Figure 10: Numerical solution and absolute error of Example 6 using $N=40$, $x_{\alpha }=0.4$ for $t\in [0,0.5]$.

Figure 11: Numerical results at final time simulation $T=0.5$ and $x_{\alpha }=0.4$ of Example 6.

Figure 12: Numerical results at final time simulation $T=0.5$ and $x_{\alpha }=0.63$ of Example 6.

Table. 6 Convergence analysis of Example 6 using the IFD-IIM for the heat equation with a general discontinuous solution.

	${\textstyle x_{\alpha }=0.4}$				${\textstyle x_{\alpha }=0.63}$
${\textstyle N}$	${\textstyle L_{\infty }}$-norm	Order	${\textstyle L_{2}}$-norm	Order	${\textstyle L_{\infty }}$-norm	Order	${\textstyle L_{2}}$-norm	Order
10	3.91e-04	–-	2.18e-04	–-	4.32e-04	–-	2.43e-04	–-
20	2.50e-05	3.97	1.34e-05	4.02	2.48e-05	4.12	1.45e-05	4.07
40	1.56e-06	4.00	8.37e-07	4.01	2.38e-06	3.38	1.12e-06	3.69
80	9.72e-08	4.00	5.22e-08	4.00	9.48e-08	4.65	5.56e-08	4.34
160	6.08e-09	4.03	3.26e-09	4.00	9.29e-09	3.35	4.37e-09	3.69
LSM		4.00		4.01		3.90		3.95

7 Conclusions

This work serves as the general foundation for addressing high-order IIMs to solve interface problems, facilitating comprehensive investigations into this theoretical framework. Here, we present a fourth-order finite-difference scheme for approximating the second-order derivative of real-valued continuous and discontinuous functions. This method combines an implicit formulation with the IIM. Our proposed scheme employs a three-point stencil, achieving different accuracy at regular and irregular points. To illustrate the effectiveness of the proposed IFD-IIM approach, we applied it to solve the one-dimensional Poisson equation and the heat conduction equation. The global accuracy of the fourth order was demonstrated using several numerical examples for both equations. Hence, this study establishes a general strategy for high-order IIMs, enabling their application to elliptic and time-evolving problems in several dimensions. Additionally, the implicit procedure lends itself to developing of higher-order numerical schemes based on the IIM, including sixth-order methods.

Acknowledgments

This work was partially supported by CONAHCYT under the program Investigadoras e Investigadores por México. We also thanks the facilities provided by the Facultad de Matemáticas (UADY) for the development of this work.

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