Derivation of Strictly Stable High Order Difference Approximations for Variable-Coefficient PDE

A new way of deriving strictly stable high order difference operators for partial differential equations (PDE) is demonstrated for the 1D convection diffusion equation with variable coefficients. The derivation is based on a diffusion term in conservative, i.e. self-adjoint, form. Fourth order accurate difference operators are constructed by mass lumping Galerkin finite element methods so that an explicit method is achieved. The analysis of the operators is confirmed by numerical tests. The operators can be extended to multi dimensions, as we demonstrate for a 2D example. The discretizations are also relevant for the Navier–Stokes equations and other initial boundary value problems that involve up to second derivatives with variable coefficients.

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

[2]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[3]  I. Stakgold.  Green's Functions and Boundary Value Problems , 1979 .

[4]  Jan Nordström,et al.  Error Bounded Schemes for Time-dependent Hyperbolic Problems , 2007, SIAM J. Sci. Comput..

[5]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[6]  A. M. Davies,et al.  The use of the galerkin method with a basis of B-Splines for the solution of the one-dimensional primitive equations , 1978 .

[7]  Antony Jameson,et al.  Formulation of Kinetic Energy Preserving Conservative Schemes for Gas Dynamics and Direct Numerical Simulation of One-Dimensional Viscous Compressible Flow in a Shock Tube Using Entropy and Kinetic Energy Preserving Schemes , 2008, J. Sci. Comput..

[8]  H. C. Yee,et al.  Entropy Splitting for High Order Numerical Simulation of Vortex Sound at Low Mach Numbers , 2001, J. Sci. Comput..

[9]  B. Gustafsson.  High Order Difference Methods for Time Dependent PDE , 2008 .

[10]  Eitan Tadmor,et al.  Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems , 2003, Acta Numerica.

[11]  B. Friedman.  Principles and Techniques of Applied Mathematics , 1956 .

[12]  A. Huerta,et al.  Finite Element Methods for Flow Problems , 2003 .

[13]  Alina Chertock,et al.  Strict Stability of High-Order Compact Implicit Finite-Difference Schemes , 2000 .

[14]  Abraham Zemui,et al.  Fourth Order Symmetric Finite Difference Schemes for the Acoustic Wave Equation , 2005 .

[15]  H. Kreiss,et al.  Time-Dependent Problems and Difference Methods , 1996 .

[16]  Magnus Svärd,et al.  On the order of accuracy for difference approximations of initial-boundary value problems , 2006, J. Comput. Phys..

[17]  G. Arfken.  Mathematical Methods for Physicists , 1967 .

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

[19]  Bertil Gustafsson,et al.  The convergence rate for difference approximations to general mixed initial boundary value problems , 1981 .

[20]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[21]  H. C. Yee,et al.  Entropy Splitting for High Order Numerical Simulation of Compressible Turbulence , 2002 .

[22]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[23]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[24]  Margot Gerritsen,et al.  Designing an efficient solution strategy for fluid flows. 1. A stable high order finite difference scheme and sharp shock resolution for the Euler equations , 1996 .

[25]  O. C. Zienkiewicz,et al.  The Finite Element Method: Its Basis and Fundamentals , 2005 .

[26]  Ken Mattsson,et al.  Boundary Procedures for Summation-by-Parts Operators , 2003, J. Sci. Comput..

[27]  J. Nordström,et al.  Summation by Parts Operators for Finite Difference Approximations of Second-Derivatives with Variable Coefficients , 2004, Journal of Scientific Computing.

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

[29]  H. Kreiss,et al.  On the stability definition of difference approximations for the initial boundary value problem , 1993 .

[30]  Jan Nordström,et al.  High-order finite difference methods, multidimensional linear problems, and curvilinear coordinates , 2001 .

[31]  J. Oden,et al.  Finite Element Methods for Flow Problems , 2003 .

[32]  A. Chertock,et al.  Strict Stability of High-Order Compact Implicit Finite-Difference Schemes: The Role of Boundary Conditions for Hyperbolic PDEs, II , 2000 .

[33]  Jan Nordström,et al.  Conservative Finite Difference Formulations, Variable Coefficients, Energy Estimates and Artificial Dissipation , 2006, J. Sci. Comput..

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