Summation-by-parts operators and high-order quadrature

Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by parts. The SBP operator definition includes a weight matrix that is used formally for discrete integration; however, the accuracy of the weight matrix as a quadrature rule is not explicitly part of the SBP definition. We show that SBP weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes. For diagonal weight matrices, the accuracy of SBP quadrature extends to curvilinear domains provided the Jacobian is approximated with the same SBP operator used for the quadrature. This quadrature has significant implications for SBP-based discretizations; in particular, the diagonal norm accurately approximates the L^2 norm for functions, and multi-dimensional SBP discretizations accurately approximate the divergence theorem.

[1]  Jianping Li.  General explicit difference formulas for numerical differentiation , 2005 .

[2]  Jason E. Hicken,et al.  Parallel Newton-Krylov Solver for the Euler Equations Discretized Using Simultaneous-Approximation Terms , 2008 .

[3]  Erik Schnetter,et al.  Optimized High-Order Derivative and Dissipation Operators Satisfying Summation by Parts, and Applications in Three-dimensional Multi-block Evolutions , 2005, J. Sci. Comput..

[4]  B. Strand.  Summation by parts for finite difference approximations for d/dx , 1994 .

[5]  Magnus Svärd,et al.  Stable and accurate schemes for the compressible Navier-Stokes equations , 2008, J. Comput. Phys..

[6]  Begnaud Francis Hildebrand,et al.  Introduction to numerical analysis: 2nd edition , 1987 .

[7]  J. Steger,et al.  Implicit Finite-Difference Simulations of Three-Dimensional Compressible Flow , 1980 .

[8]  Magnus Svärd,et al.  Stable and Accurate Artificial Dissipation , 2004, J. Sci. Comput..

[9]  B. Gustafsson.  The convergence rate for difference approximations to mixed initial boundary value problems , 1975 .

[10]  P. Thomas,et al.  Geometric Conservation Law and Its Application to Flow Computations on Moving Grids , 1979 .

[11]  D. Gottlieb,et al.  Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes , 1994 .

[12]  Manuel Tiglio,et al.  Orbiting binary black hole evolutions with a multipatch high order finite-difference approach , 2009, 0904.0493.

[13]  F. B. Hildebrand.  Introduction to Numerical Analysis , 1976 .

[14]  David W. Zingg,et al.  A parallel Newton-Krylov-Schur flow solver for the Navier-Stokes equations using the SBP-SAT approach , 2010 .

[15]  Magnus Svärd,et al.  On Coordinate Transformations for Summation-by-Parts Operators , 2004, J. Sci. Comput..

[16]  Jason E. Hicken,et al.  Superconvergent Functional Estimates from Summation-By-Parts Finite-Difference Discretizations , 2011, SIAM J. Sci. Comput..

[17]  H. Kreiss,et al.  Finite Element and Finite Difference Methods for Hyperbolic Partial Differential Equations , 1974 .

[18]  A. M. Kuethe,et al.  Foundations of aerodynamics: bases of aerodynamic design , 1986 .

[19]  Jing Gong,et al.  A stable and conservative high order multi-block method for the compressible Navier-Stokes equations , 2009, J. Comput. Phys..

[20]  D. Gottlieb,et al.  A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations , 1988 .