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A priori error estimates of an extrapolated space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. (English) Zbl 1237.65105
The numerical solution of a scalar nonstationary nonlinear convection-diffusion equation for 2D or 3D problems is studied. The aim is to develop a sufficiently efficient, robust and accurate numerical scheme for the simulation of unsteady viscous compressible flows. For the numerical scheme a combination of the discontinuous Galerkin finite element method for the space as well for the time discretization is used. In the numerical scheme, the linear diffusive and penalty terms are computed implicitly and the nonlinear convective term is treated in the previous time step. For this purpose a special higher order explicit extrapolation is used. A priori asymptotic error estimates in $$L_\infty((0,T), L_2(\Omega))$$ and $$L_2((0,T), H^1(\Omega))$$ functional spaces under some relation on the mesh size $$h$$ and the time step $$\tau$$ are proved. Several numerical experiments together with the implementation notes for the proposed scheme conclude the paper. These numerical experiments verify the theoretical results.

MSC:
 65M15 Error bounds 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 35K55 Nonlinear parabolic equations 76N15 Gas dynamics (general theory) 76M10 Finite element methods applied to problems in fluid mechanics
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References:
 [1] Bassi, Discontinuous Galerkin solution of the Reynolds averaged Navier-Stokes and k - {$$\omega$$} turbulence model equations, Comput Fluids 34 pp 507– (2005) · Zbl 1138.76043 [2] Dolejší, Semi-implicit interior penalty discontinuous Galerkin methods for viscous compressible flows, Commun Comput Phys 4 pp 231– (2008) [3] Dumbser, Building blocks for arbitrary high-order discontinuous Galerkin methods, J Sci Comput 27 pp 215– (2006) · Zbl 1115.65100 [4] Ern, A well-balanced Runge-Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying, Int J Numer Methods Fluids 58 pp 1– (2008) · Zbl 1142.76036 [5] Hartmann, Symmetric interior penalty DG methods for the compressible Navier-Stokes equations I: Method formulation, Int J Numer Anal Model 1 pp 1– (2006) · Zbl 1129.76030 [6] Klaij, Pseudo-time stepping for space-time discontinuous Galerkin discretizations of the compressible Navier-Stokes equations, J Comput Phys 219 pp 622– (2006) · Zbl 1102.76035 [7] Arnold, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J Numer Anal 39 pp 1749– (2002) · Zbl 1008.65080 [8] Dolejší, On the discontinuous Galerkin method for the numerical solution of the Navier-Stokes equations, Int J Numer Methods Fluids 45 pp 1083– (2004) · Zbl 1060.76570 [9] Gear, Numerical initial value problems in ordinary differential equations (1971) · Zbl 1145.65316 [10] Hairer, Springer Series in Computational Mathematics, no. 8, in: Solving ordinary differential equations i, nonstiff problems (2000) [11] Hairer, Solving ordinary differential equations ii, stiff and differential-algebraic problems (2002) · Zbl 0859.65067 [12] Crouzeix, Une méthode multipas implicit-explicit pour l’approximation des équations d’évolutions paraboliques, Numer Math 35 pp 257– (1980) · Zbl 0419.65057 [13] Varah, Stability restrictions on second order, three level finite difference scheme for parabolic equations, SIAM J Numer Anal 17 pp 300– (1980) · Zbl 0426.65048 [14] Dolejší, Analysis of a BDF-DGFE scheme for nonlinear convection-diffusion problems, Numer Math 110 pp 405– (2008) · Zbl 1158.65068 [15] Sochala, Mass conservative BDF-discontinuous Galerkin/explicit finite volume schemes for coupling subsurface and overland flows, Comput Methods Appl Mech Eng 198 pp 2122– (2009) · Zbl 1227.76045 [16] Eriksson, Computational differential equations (1996) [17] Thomée, Galerkin finite element methods for parabolic problems. (2006) · Zbl 1105.65102 [18] Chrysafinos, Error estimates for discontinuous Galerkin approximations of implicit parabolic equations, SIAM J Numer Anal 43 pp 2478– (2006) · Zbl 1110.65088 [19] Chrysafinos, Error estimates for the discontinuous Galerkin methods for parabolic equations, SIAM J Numer Anal 44 pp 349– (2006) · Zbl 1112.65086 [20] Jamet, Galerkin-type approximations which are discontinuous in time for parabolic equation in a variable domain, SIAM J Numer Anal 15 pp 912– (1978) · Zbl 0434.65091 [21] Luskin, On the smoothing property of the galerkin method for parabolic equations, SIAM J Numer Anal 19 pp 93– (1982) · Zbl 0483.65064 [22] D. Schötzau hp-DGFEM for parabolic evolution problems. Applications to diffusion and viscous incompressible flow 1999 [23] Schötzau, An hp a priori error analysis of the discontinuous galerkin time-stepping for initial value problems, Calcolo 37 pp 207– (2000) · Zbl 1012.65084 [24] Werder, hp-discontinuous Galerkin time stepping for parabolic problems, Comput Methods Appl Mech Eng 190 pp 6685– (2001) · Zbl 0992.65103 [25] Jaffre, Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws, Math Models Methods Appl Sci 5 pp 367– (1995) · Zbl 0834.65089 [26] Rivière, Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation, Frontiers in Applied Mathematics (2008) · Zbl 1153.65112 [27] Sudirham, Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains, Appl Numer Math 56 pp 1491– (2006) · Zbl 1111.65089 [28] Lörcher, A discontinuous Galerkin scheme based on a spacetime expansion. I. Inviscid compressible flow in one space dimension, J Sci Comput 32 pp 175– (2007) · Zbl 1143.76047 [29] Dolejší, Analysis of the discontinuous galerkin method for nonlinear convection-diffusion problems, Comput Methods Appl Mech Eng 194 pp 2709– (2005) · Zbl 1093.76034 [30] Feistauer, Space-time discontinuos Galerkin method for solving nonstationary convection-diffusion-reaction problems, Appl Math Praha 52 pp 197– (2007) · Zbl 1164.65469 [31] Nečas, Les méthodes directes en thèorie des equations elliptiques (1967) [32] Lions, Mathematical topics in fluid mechanics (1996) [33] Feistauer, Mathematical and Computational Methods for Compressible Flow (2003) [34] Agmon, Lectures on elliptic boundary value problems (1965) [35] Dolejší, A finite volume discontinuous Galerkin scheme for nonlinear convection-diffusion problems, Calcolo 39 pp 1– (2002) · Zbl 1098.65095 [36] Ciarlet, The finite elements method for elliptic problems (1979) [37] Brenner, The mathematical theory of finite element methods (1994) · Zbl 0804.65101 [38] Watkins, Wiley-Interscience Series of Texts, Monographs, and Tracts, in: Fundamentals of matrix computations, Pure and Applied Mathematics (2002)
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