Published in Comput. Methods Appl. Mech. Engrg. Vol. 199, pp. 525–546, 2010
doi: 10.1016/j.cma.2009.10.009

Abstract

We present the design of a high-resolution Petrov–Galerkin (HRPG) method using linear finite elements for the problem defined by the residual $R(\phi ):=\partial \phi /\partial t+u\partial \phi /\partial x-k\partial ^{2}\phi /\partial x^{2}+s\phi -f$ where $k;\geq s0$. The structure of the method in 1D is identical to the consistent approximate upwind Petrov–
Galerkin (CAU/PG) method [A.C. Galeão, E.G. Dutra do Carmo, A consistent approximate upwind Petrov–Galerkin method for the convection-dominated problems, Comput. Methods Appl. Mech. Engrg. 68(1988) 83–95] except for the definitions of the stabilization parameters. Such a structure may also be attained via the finite-calculus (FIC) procedure [E. Oñate, Derivation of stabilized equations for numerical solution of advective–diffusive transport and fluid flow problems, Comput. Methods Appl. Mech. Engrg. 151 (1998) 233–265; E. Oñate, J. Miquel, G. Hauke, Stabilized formulation for the advection–diffusion–
absorption equation using finite-calculus and linear finite elements, Comput. Methods Appl. Mech. Engrg. 195 (2006) 3926–3946] by an appropriate definition of the characteristic length. The prefix ‘high-resolution’ is used here in the sense popularized by Harten, i.e. second order accuracy for smooth/regular regimes
and good shock-capturing in nonregular regimes. The design procedure embarks on the problem of circumventing the Gibbs phenomenon observed in L2-projections. Next we study the conditions on
the stabilization parameters to circumvent the global oscillations due to the convective term. A conjuncture of the two results is made to deal with the problem at hand that is usually plagued by Gibbs, global
and dispersive oscillations in the numerical solution. It is shown that the method indeed reproduces stabilized high-resolution numerical solutions for a wide range of values of $u;k;s$ and $f$. Finally, some remarks are made on the extension of the HRPG method to multidimensions.

S. Siryk. Accuracy and Stability of the Petrov–Galerkin Method for Solving the Stationary Convection-Diffusion Equation. Cybern Syst Anal 50(2) (2014) DOI 10.1007/s10559-014-9615-7

Z. Jingbo, Z. Xun, C. Ruige, S. Chao, L. Yaping, W. Xiaocui. Influence of variation in temperature on groundwater flow nets in a geothermal system. Environ Earth Sci 73(3) (2014) DOI 10.1007/s12665-014-3453-9

J. Zhao, X. Zhou, Y. Yu, C. Ruan, X. Wang, J. Li, Q. Zhang, X. Shen. Changes in fluid flux and hydraulic head in a geothermal confined aquifer. Environ Earth Sci 75(5) (2016) DOI 10.1007/s12665-016-5248-7

S. Shafaq, A. Kennedy, Y. Delauré. A method for numerical modelling of convection-reaction-diffusion using electrical analogues. Int. J. Numer. Model. 29(3) (2015) DOI 10.1002/jnm.2091

S. Arora, I. Kaur, F. Potůček. MODELLING OF DISPLACEMENT WASHING OF PULP FIBERS USING THE HERMITE COLLOCATION METHOD. Braz. J. Chem. Eng. 32(2) DOI 10.1590/0104-6632.20150322s00003160

F. Mossaiby, R. Rossi, P. Dadvand, S. Idelsohn. OpenCL-based implementation of an unstructured edge-based finite element convection-diffusion solver on graphics hardware. Int. J. Numer. Meth. Engng 89(13) (2011) DOI 10.1002/nme.3302