Switch flux limiter method for viscous and nonviscous conservation laws Silvia Jerez ⇑ , Mario Arciga Department of Applied Mathematics, CIMAT, Guanajuato, Gto. 36240, Mexico article info Keywords: Viscous flux limiter methods Viscous conservation laws FORWE method abstract In this work we develop an efficient shock capturing scheme of the TVD flux limiter family for viscous and nonviscous conservation laws. The new flux limiter method is based on the monotone FORWE scheme which is optimized by the inclusion of an appropriate switch function. For the viscous case, a conservative formulation of the type viscous flux limiter defined by Toro is used. Theoretical properties such as nonlinear stability and weak conver- gence are proven using TVD-stability. An efficiency analysis of the method is achieved by validating the numerical results with the analytical solutions of benchmark nonviscous and viscous problems. We compare the switch flux limiter results with those obtained by some of the well known flux limiter methods. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction We consider the scalar viscous conservation law @uðx; tÞ @t þ r  hðuðx; tÞÞ ¼ 0; x 2 X; t 2½0; T ; ð1Þ uðx; 0Þ¼ u 0 ðxÞ; where the physical flux is given by the Fourier–Fick’s law as follows hðuÞ¼ f ðuÞ rlðuÞ; ð2Þ with X a bounded domain such that, x 2 X # R m ; u ¼ uðx; tÞ a conservative scalar function with advective vectorial flux f ¼ f ðuðx; tÞÞ and diffusion flux l ¼ lðuðx; tÞÞ with lðuÞ P 0. These parabolic equations often arise in real-life applications like front propagation, reservoir simulation in porous media, and so on [1,2]. If lðuÞ¼ 0, Eq. (1) becomes the hyperbolic problem @uðx; tÞ @t þ r  f ðuðx; tÞÞ ¼ 0; ð3Þ which has discontinuous solutions. So the existence of classical solutions can not be guaranteed even for smooth initial con- ditions. Let uðx; tÞ be a weak solution of the problem (3), then to achieve uniqueness of solution is required an entropy con- dition like the one given by Volpert and Kruz ˇkov in [3,4]. Analogous difficulties appear for a viscous conservation law with http://dx.doi.org/10.1016/j.amc.2014.08.011 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail address: jerez@cimat.mx (S. Jerez). Applied Mathematics and Computation 246 (2014) 292–305 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc