Abstract

We revisit the theory of Discrete Exterior Calculus (DEC) in 2D for general triangulations, relying only on Vector Calculus and Matrix Algebra. We present DEC numerical solutions of the Poisson equation and compare them against those found using the Finite Element Method with linear elements (FEML).

Abstract

Graphic processing units have received much attention in last years. Compute-intensive algorithms operating on multidimensional arrays that have nearest neighbor dependency and/or exploit data locality can achieve massive speedups. Simulation of problems modeled by time-dependent Partial Differential Equations by using explicit time-stepping methods on structured grids is an instance of such GPU-friendly algorithms. Solvers for transient incompressible fluid flow cannot be developed in a fully explicit manner due to the incompressibility constraint. Segregated algorithms like the fractional step method require the solution of a Poisson problem for the pressure field at each time level. This stage is usually the most time-consuming one. This work discuss a solver for the pressure problem in applications using immersed boundary techniques in order to account for moving solid bodies. This solver is based on standard Conjugate Gradients iterations and depends on the availability of a fast Poisson solver on the whole domain to define a preconditioner. We provide a theoretical and numerical evidence on the advantages of our approach versus classical techniques based on fixed point iterations such as the Iterated Orthogonal Projection method.

Abstract

We present a local formulation for 2D Discrete Exterior Calculus (DEC) similar to that of the Finite Element Method (FEM), which allows a natural treatment of material heterogeneity (assigning material properties element by element). It also allows us to deduce, in a principled manner, anisotropic fluxes and the DEC discretization of the pullback of 1-forms by the anisotropy tensor, i.e. we deduce the the discrete action of the anisotropy tensor on primal 1-forms. Due to the local formulation, the computational cost of DEC is similar to that of the Finite Element Method with Linear interpolating functions (FEML). The numerical DEC solutions to the anisotropic Poisson equation show numerical convergence, are very close to those of FEML on fine meshes and are slightly better than those of FEML on coarse meshes.

Abstract

Abstract

This paper introduces a sixth-order Immersed Interface Method (IIM) for addressing 2D Poisson problems characterized by a discontinuous forcing function with straight interfaces. In the presence of this discontinuity, the problem exhibits a non-smooth solution at the interface that divides the domain into two regions. Here, the IIM is employed to compute the solution on a fixed Cartesian grid. This method integrates necessary jump conditions resulting from the interface into the numerical schemes. In order to achieve a sixth-order method, the proposed approach combines implicit finite differences with the IIM. The proposed scheme is efficient because the matrix arising from discretization remains the same as in the smooth problem, and changes are made to the resulting linear system by introducing new terms on the right side. These supplementary terms account for the discontinuities in the solution and its derivatives, with calculations restricted near the interface. The paper demonstrates the accuracy of the proposed method through various numerical examples.

Abstract

This work forms the foundation for addressing high-order immersed interface methods to solve interface problems and enables us to conduct in-depth examination of this theory. Here, we focus on the introduction a fourth-order finite-difference formulation to approximate the second-order derivative of discontinuous functions. The approach is based on the combination of a high-order implicit formulation and the immersed interface method. The idea is to modify the standard schemes by introducing additional contribution terms based on jump conditions. These contributions are calculated only at grid points where the stencil intersects with the interface. Here, we discuss the issues of implementing the one-dimensional Poisson equation and the heat conduction equation with discontinuous solutions as a three-point stencil for each grid point on the computational domain. In both cases, the resulting discretization approach yields a tridiagonal linear system with matrix coefficients identical to those employed for smooth solutions. We present several numerical experiments to verify the feasibility and accuracy of the method. Thus, this high-order method provides an attractive numerical framework that can efficiently lead to the solution to more complex problems.

Abstract

Abstract

Discrete versions of Poisson's equation with large contrasts in the coefficients result in very ill-conditioned systems. Thus, its iterative solution represents a major challenge, for instance, in porous media and multiphase flow simulations, where considerable permeability and density ratios are usually found. The existing strategies trying to remedy this are highly dependent on whether the coefficient matrix remains constant at each time iteration or not. In this regard, incompressible multiphase flows with high-density ratios are particularly demanding as their resulting Poisson equation varies along with the density field, making the reconstruction of complex preconditioners impractical. This work presents a strategy for solving such versions of the variable Poisson equation.Roughly, we first make it constant through an adequate approximation. Then, we block-diagonalise it through an inexpensive change of basis that takes advantage of mesh reflection symmetries, which are common in multiphase flows. Finally, we solve the resulting set of fully decoupled subsystems with virtually any solver. The numerical experiments conducted on a multiphase flow simulation prove the benefits of such an approach, resulting in up to 6.6x faster convergences.