×

zbMATH — the first resource for mathematics

Error estimates to smooth solutions of semi-discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws. (English) Zbl 1361.65067
This paper deals with error estimates to smooth solutions of semi-discrete discontinuous Galerkin (DG) methods with the \(P^k\) finite element space of piecewise \(k\)th degree polynomials and quadrature rules for scalar conservation laws. In the case of 1D and under the assumption that the exact solution and the physical flux are smooth, it is shown that the error estimate is of order \(O(h^{k+\frac{1}{2}})\) for general monotone fluxes while the order is \(O(h^{k+1})\) for upwind fluxes if the quadrature over elements is exact for polynomials of degree \((2k)\). For multidimensional problems, it is proved that, if the quadrature over elements is exact for polynomials of degree \((2k+1)\), the error estimate is of order \(O(h^{k+\frac{1}{2}})\) for smooth numerical fluxes. The techniques presented are based on the use of the energy estimate and Taylor expansions first introduced by Q. Zhang and C.-W. Shu [SIAM J. Numer. Anal. 42, No. 2, 641–666 (2004; Zbl 1078.65080)]. Some numerical results are presented to support the theoretical results.

MSC:
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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
35J65 Nonlinear boundary value problems for linear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Reed, Triangular mesh methods for the neutron transport equation, Los Alamos Report (1973)
[2] Cockburn, The Runge-Kutta local projection P1-discontinuous-Galerkin finite element method for scalar conservation laws, ESAIM: Math Model Num Anal 25 pp 337– (1991) · Zbl 0732.65094
[3] Cockburn, TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II. General framework, Math Comput 52 pp 411– (1989)
[4] Cockburn, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J Comput Phys 84 pp 90– (1989) · Zbl 0677.65093
[5] Cockburn, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Math Comput 54 pp 545– (1990) · Zbl 0695.65066
[6] Cockburn, The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J Comput Phys 141 pp 199– (1998) · Zbl 0920.65059
[7] Shu, Total-variation-diminishing time discretizations, SIAM J Sci Stat Comput 9 pp 1073– (1988) · Zbl 0662.65081
[8] Shu, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J Comput Phys 77 pp 439– (1988) · Zbl 0653.65072
[9] Shu, Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J Comput Phys 83 pp 32– (1989) · Zbl 0674.65061
[10] Cockburn, Advanced numerical approximation of nonlinear hyperbolic equations pp 151– (1998) · Zbl 0904.00047
[11] Cockburn, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J Sci Comput 16 pp 173– (2001) · Zbl 1065.76135
[12] P. Lesaint P.-A. Raviart On a finite element method for solving the neutron transport equation, Mathematical Aspects Finite Elements Partial Differential Equations 33 1974 89 123
[13] Johnson, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math Comput 46 pp 1– (1986) · Zbl 0618.65105
[14] Richter, An optimal-order error estimate for the discontinuous Galerkin method, Math Comput 50 pp 75– (1988) · Zbl 0643.65059
[15] Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM J Num Anal 28 pp 133– (1991) · Zbl 0729.65085
[16] Cockburn, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J Num Anal 35 pp 2440– (1998) · Zbl 0927.65118
[17] Zhang, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J Num Anal 42 pp 641– (2004) · Zbl 1078.65080
[18] Zhang, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws, SIAM J Num Anal 44 pp 1703– (2006) · Zbl 1129.65062
[19] Zhang, Stability analysis and a priori error estimates of the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws, SIAM J Num Anal 48 pp 1038– (2010) · Zbl 1217.65178
[20] Sobotíková, Effect of numerical integration in the DGFEM for nonlinear convection-diffusion problems, Num Methods Partial Differential Equations 23 pp 1368– (2007) · Zbl 1133.65079
[21] Sobotíková, Numerical integration in the DGFEM for 3D nonlinear convection-diffusion problems on nonconforming meshes, Num Funct Anal Opt 29 pp 927– (2008) · Zbl 1156.65086
[22] Yan, A local discontinuous Galerkin method for KdV type equations, SIAM J Num Anal 40 pp 769– (2002) · Zbl 1021.65050
[23] Dolejší, A finite volume discontinuous Galerkin scheme for nonlinear convection-diffusion problems, Calcolo 39 (1) pp 1– (2002) · Zbl 1098.65095
[24] P. G. Ciarlet The finite element method for elliptic problems 1978
[25] Kučera, On diffusion-uniform error estimates for the DG method applied to singularly perturbed problems, IMA J Num Anal 34 pp 820– (2014) · Zbl 1305.65203
[26] Harten, High resolution schemes for hyperbolic conservation laws, J Comput Phys 49 pp 357– (1983) · Zbl 0565.65050
[27] Dunavant, High degree efficient symmetrical Gaussian quadrature rules for the triangle, Int J Num Eng 21 pp 1129– (1985) · Zbl 0589.65021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.