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

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Abstract

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.

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References

  1. Abarbanel, S.S., Chertock, A.: Strict stability of high-order compact implicit finite-difference schemes: the role of boundary conditions for hyperbolic PDEs, I. J. Comput. Phys. 160(1), 42–66 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abarbanel, S.S., Chertock, A., Yefet, A.: Strict stability of high-order compact implicit finite-difference schemes: the role of boundary conditions for hyperbolic PDEs, II. J. Comput. Phys. 160(1), 67–87 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arfken, G., Weber, H.: Mathematical Methods for Physicists. Academic Press, London (2001)

    MATH  Google Scholar 

  4. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Carpenter, M., Gottlieb, D., Abarbanel, S.: Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to high-order compact schemes. J. Comput. Phys. 111(2), 220–236 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  6. Davis, A.: The use of the Galerkin method with a basis of B-splines for the solution of the one-dimensional primitive equation. J. Comput. Phys. 27, 123–137 (1976)

    Article  Google Scholar 

  7. Donea, J., Huerta, A.: Finite Element Methods for Flow Problems. Wiley, Chichester (2003)

    Book  Google Scholar 

  8. Friedman, B.: Principles and Techniques of Applied Mathematics. Wiley, New York (1966)

    Google Scholar 

  9. Gerritsen, M., Olsson, P.: Designing an efficient solution strategy for fluid flows 1. A stable high order finite difference scheme and sharp shock resolution for the Euler equations. J. Comput. Phys. 129, 245–262 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gustafsson, B.: The convergence rate for difference approximations to general mixed initial boundary value problems. SIAM J. Numer. Anal. 18(2), 179–190 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gustafsson, B.: High Order Difference Methods for Time Dependent PDE. Springer, Berlin (2008)

    MATH  Google Scholar 

  12. Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time Dependent Problems and Difference Methods. Wiley, New York (1995)

    MATH  Google Scholar 

  13. Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd edn. Springer, Berlin (1993)

    MATH  Google Scholar 

  14. Hughes, T.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover, New York (2000)

    MATH  Google Scholar 

  15. Jameson, A.: 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. J. Sci. Comput. 34, 188–208 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kormann, K., Kronbichler, M.: High order finite difference approximations for parabolic and hyperbolic-parabolic problems with variable coefficients. Project Report, Division of Scientific Computing, Department of Information Technology, Uppsala University (2006). Available online: http://user.it.uu.se/~martinkr/kormann-kronbichler_project.pdf

  17. Kreiss, H.-O., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press, New York (1974)

    Google Scholar 

  18. Kreiss, H.-O., Wu, L.: On the stability definition of difference approximations for the initial boundary value problem. Appl. Numer. Math. 12, 213–227 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  19. Mattsson, K.: Boundary procedures for summation-by-parts operators. J. Sci. Comput. 18, 133–153 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Mattsson, K., Nordström, J.: Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199, 503–540 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Müller, B., Yee, H.: Entropy splitting for high order numerical simulation of vortex sound at low Mach numbers. J. Sci. Comput. 17, 181–190 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Nordström, J.: Conservative finite difference formulations, variable coefficients, energy estimates and artificial dissipation. J. Sci. Comput. 29(3), 375–404 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Nordström, J.: Error bounded schemes for time-dependent hyperbolic problems. SIAM J. Sci. Comput. 30(1), 46–59 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Nordström, J., Carpenter, M.: High-order finite difference methods, multidimensional linear problems, and curvilinear coordinates. J. Comput. Phys. 199, 503–540 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Sandham, N., Li, O., Yee, H.: Entropy splitting for high-order numerical simulation of compressible turbulence. J. Comput. Phys. 178, 307–322 (2002)

    Article  MATH  Google Scholar 

  26. Stakgold, I.: Green’s Functions and Boundary Value Problems. Wiley, New York (1998)

    MATH  Google Scholar 

  27. Strand, B.: Summation by parts for finite difference approximations for d/dx. J. Comput. Phys. 110, 47–67 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  28. Strang, G., Fix, G.: An Analysis of the Finite Element Method. Series in Automatic Computation. Prentice-Hall, Englewood Cliffs (1973)

    MATH  Google Scholar 

  29. Svärd, M.: On coordinate transformations for summation-by-parts operators. J. Sci. Comput. 20(1), 29–42 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  30. Svärd, M., Nordström, J.: On the order of accuracy for difference approximations of initial-boundary value problems. J. Comput. Phys. 218(1), 333–352 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  32. Zemui, A.: High order symmetric finite difference schemes for the acoustic wave equation. PhD thesis, Faculty of Science and Technology, Uppsala University (2003)

  33. Zienkiewicz, O., Taylor, R., Zhu, J.: The Finite Element Method: Its Basis and Fundamentals, 6th edn. Elsevier, Amsterdam (2005)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Division of Scientific Computing, Department of Information Technology, Uppsala University, Uppsala, Sweden

    Katharina Kormann & Martin Kronbichler

  2. Department of Energy and Process Engineering, Norwegian University of Science and Technology (NTNU), Trondheim, Norway

    Bernhard Müller

Authors
  1. Katharina Kormann
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  2. Martin Kronbichler
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  3. Bernhard Müller
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Correspondence to Katharina Kormann.

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Kormann, K., Kronbichler, M. & Müller, B. Derivation of Strictly Stable High Order Difference Approximations for Variable-Coefficient PDE. J Sci Comput 50, 167–197 (2012). https://doi.org/10.1007/s10915-011-9479-1

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  • DOI: https://doi.org/10.1007/s10915-011-9479-1

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