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Numerical solutions of the 1D convection-diffusion-reaction and the Burgers equation using implicit multi-stage and finite element methods. (English) Zbl 1279.65107
Constanda, Christian (ed.) et al., Integral methods in science and engineering. Progress in numerical and analytic techniques. Proceedings of the conference, IMSE, Bento Gonçalves, Rio Grande do Sul, Brazil, July 23–27, 2012. New York, NY: Birkhäuser/Springer (ISBN 978-1-4614-7827-0/hbk; 978-1-4614-7828-7/ebook). 205-216 (2013).
Summary: We apply the semi-discrete formulation, where the time variable is discretized using an implicit multi-stage method and the space variable is discretized using the finite element method, to obtain numerical solutions for the 1D convection-diffusion-reaction and the Burgers equation, whose analytical solutions are known. More specifically, we use the implicit multi-stage method of second- and fourth-order for time discretization. For space discretization, we use three finite elements methods, least squares (LSFEM), Galerkin (GFEM) and streamline-upwind Petrov-Galerkin (SUPG). We present an error analysis, comparing the numerical with analytical solutions. We verify that the implicit multi-stage second-order method when combined with the LSFEM, GFEM and SUPG, increases the region of convergence of the numerical solutions. LSFEM presents the better performance when compared to GFEM and SUPG.
For the entire collection see [Zbl 1275.00035].

MSC:
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35Q53 KdV equations (Korteweg-de Vries equations)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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