×

Summation by parts operators for finite difference approximations of second derivatives. (English) Zbl 1071.65025

J. Comput. Phys. 199, No. 2, 503-540 (2004); erratum ibid. 351, 535 (2017).
The authors design central difference operators to approximate the derivatives \(d/dx\), \(d^2/dx^2\) at evenly spaced points in an interval \([a,b]\) which satisfy numerical formulas analogous to the integration by parts formulas from calculus (summation by parts (SBP) formulas). Using discrete diagonal norms the authors construct third, fourth, and fifth degree accurate SBP operators approximating \(d/dx\), \(d^2/dx^2\) and using discrete matrix norms fourth, sixth, and eighth accurate approximations. These approximations lead to easily obtained energy estimates for mixed hyperbolic parabolic problems in \((x,t)\) variables provided that the initial-boundary conditions are imposed such that the SBP property is preserved. For this purpose the authors employ a method referred to as the SAT method for boundary conditions [M. Carpenter, D. Gottlieb and S. Abarbanel, J. Comput. Phys. 111, No. 2, 220–236 (1994; Zbl 0832.65098)].
Energy estimates and an error analysis are made for the convection diffusion equation and a model for the compressible Navier-Stokes equation. To obtain \(2p\)-order of accuracy it is proved that the approximating second derivatives should have order \(2p\) in the interior region and \(2p-2\) near the boundary.

MSC:

65D25 Numerical differentiation
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35M10 PDEs of mixed type
35Q30 Navier-Stokes equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Citations:

Zbl 0832.65098

Software:

Maple
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abarbanel, S.; Ditkowski, A., Asymptotically stable fourth-order accurate schemes for the diffusion equation on complex shapes, J. Comput. Phys., 133, 279-288 (1997) · Zbl 0891.65099
[2] S. Abarbanel, A. Ditkowski, A. Yefet, Bounded error schemes for the wave equation on complex domains, Technical Report 98-50, ICASE, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1998; S. Abarbanel, A. Ditkowski, A. Yefet, Bounded error schemes for the wave equation on complex domains, Technical Report 98-50, ICASE, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1998 · Zbl 1092.65069
[3] Carpenter, Mark H.; Gottlieb, David; Abarbanel, Saul, The stability of numerical boundary treatments for compact high-order finite-difference schemes, J. Comput. Phys., 108, 2 (1994) · Zbl 0832.65098
[4] Carpenter, Mark H.; Nordström, Jan; Gottlieb, David, A stable and conservative interface treatment of arbitrary spatial accuracy, J. Comput. Phys., 148 (1999) · Zbl 0921.65059
[5] Gerritsen, M.; Olsson, P., Designing an efficient solution strategy for fluid flows, J. Comput. Phys., 129, 245-262 (1996) · Zbl 0899.76281
[6] Golub, G. H.; Van Loan, C. F., Matrix Computations (1989), Johns Hopkins University Press: Johns Hopkins University Press Baltimore · Zbl 0733.65016
[7] Gustafsson, B.; Olsson, P., Fourth-order difference methods for hyperbolic IBVPs, J. Comput. Phys., 117, 1 (1995) · Zbl 0823.65083
[8] Gustafsson, Bertil, The convergence rate for difference approximations to general mixed initial boundary value problems, SIAM J. Numer. Anal., 18, 2, 179-190 (1981) · Zbl 0469.65068
[9] Gustafsson, Bertil; Kreiss, Heinz-Otto; Oliger, Joseph, Time Dependent Problems and Difference Methods (1995), Wiley: Wiley New York · Zbl 0843.65061
[10] Harten, A., The artificial compression method for computation of shocks and contact discontinuities: self adjusting hybrid schemes, Math. Comp., 32, 363-389 (1978) · Zbl 0409.76057
[11] Kreiss, H.-O.; Scherer, G., Finite element and finite difference methods for hyperbolic partial differential equations, Mathematical Aspects of Finite Elements in Partial Differential Equations (1974), Academic Press: Academic Press New York · Zbl 0355.65085
[12] H.-O. Kreiss, G. Scherer, On the existence of energy estimates for difference approximations for hyperbolic systems, Technical Report, Department of Scientific Computing, Uppsala University, 1977; H.-O. Kreiss, G. Scherer, On the existence of energy estimates for difference approximations for hyperbolic systems, Technical Report, Department of Scientific Computing, Uppsala University, 1977
[13] Mattsson, Ken, Boundary procedures for summation-by-parts operators, J. Scientific Comput., 18 (2003) · Zbl 1024.76031
[14] Ken Mattsson, Jan Nordström, Finite difference approximations of second derivatives on summation by parts form, Technical Report 2003-012, Department of Information Technology, Uppsala University, 2003. Available from <http://www.it.uu.se/research/reports/2003-012/; Ken Mattsson, Jan Nordström, Finite difference approximations of second derivatives on summation by parts form, Technical Report 2003-012, Department of Information Technology, Uppsala University, 2003. Available from <http://www.it.uu.se/research/reports/2003-012/
[15] Mattsson, Ken; Svärd, Magnus; Nordström, Jan, Stable and accurate artificial dissipation, J. Scientific Comput., 21, 1 (2004), (in press) · Zbl 1085.76050
[16] Müller, B.; Yee, H. C., High order numerical simulation of sound generated by the Kirchhoff vortex, Computing Visualization Sci., 4, 197-204 (2002) · Zbl 1009.76066
[17] Nordström, J., The use of characteristic boundary conditions for the Navier-Stokes equations, Comput. Fluids, 24, 609-623 (1995) · Zbl 0845.76075
[18] Olsson, Pelle, Summation by parts projections and stability I, Math. Comput., 64, 1035 (1995) · Zbl 0828.65111
[19] Olsson, Pelle, Summation by parts projections and stability II, Math. Comput., 64, 1473 (1995) · Zbl 0848.65064
[20] B. Sjogreen, Multiresolution wavelet based adaptive numerical dissipation control for shock-turbulence computations, Technical Report 01.01, RIACS, NASA Ames Research Center, 2000; B. Sjogreen, Multiresolution wavelet based adaptive numerical dissipation control for shock-turbulence computations, Technical Report 01.01, RIACS, NASA Ames Research Center, 2000
[21] Strand, Bo, Summation by parts for finite difference approximations for d/d \(x\), J. Comput. Phys., 110, 47-67 (1994) · Zbl 0792.65011
[22] Bo Strand, High-order difference approximations for hyperbolic initial boundary value problems, Ph.D. Thesis, Uppsala University, Department of Scientific Computing, Information Technology Uppsala University, Uppsala, Sweden, 1996; Bo Strand, High-order difference approximations for hyperbolic initial boundary value problems, Ph.D. Thesis, Uppsala University, Department of Scientific Computing, Information Technology Uppsala University, Uppsala, Sweden, 1996 · Zbl 0863.65049
[23] Yee, H. C.; Sandham, N. D.; Djomehri, M. J., Low-dissipative high-order shock-capturing methods using characteristic based filters, J. Comput. Phys., 150, 199-238 (1999) · Zbl 0936.76060
[24] H.C. Yee, M. Vinokur, M.J. Djomehri. Entropy splitting and numerical dissipation, NASA/TM-1999-208793 and 8th International Symposium on CFD, September 5-10, 1999, Bremen, Germany; H.C. Yee, M. Vinokur, M.J. Djomehri. Entropy splitting and numerical dissipation, NASA/TM-1999-208793 and 8th International Symposium on CFD, September 5-10, 1999, Bremen, Germany · Zbl 0987.65086
[25] D.W. Zingg, S. De Rango, M. Nemec, T.H. Pulliam, Comparison of several spatial discretizations for the Navier-Stokes equations, AIAA Paper 99-3260, 1999; D.W. Zingg, S. De Rango, M. Nemec, T.H. Pulliam, Comparison of several spatial discretizations for the Navier-Stokes equations, AIAA Paper 99-3260, 1999 · Zbl 0967.76073
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.