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l 1-error estimates on the immersed interface upwind scheme for linear convection equations with piecewise constant coefficients: A simple proof

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Abstract

A linear convection equation with discontinuous coefficients arises in wave propagation through interfaces. An interface condition is needed at the interface to select a unique solution. An upwind scheme that builds this interface condition into its numerical flux is called the immersed interface upwind scheme. An l1-error estimate of such a scheme was first established by Wen et al. (2008). In this paper, we provide a simple analysis on the l 1-error estimate. The main idea is to formulate the solution to the underline initial-value problem into the sum of solutions to two convection equations with constant coefficients, which can then be estimated using classical methods for the initial or boundary value problems.

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References

  1. Ambrosio L. Transport equation and Cauchy problem for BV vector fields. Invent Math, 2004, 158: 227–260

    Article  MATH  MathSciNet  Google Scholar 

  2. Bouchut F. Renormalized solutions to the Vlasov equation with coefficients of bounded variations. Arch Rat Mech Anal, 2001, 157: 75–90

    Article  MATH  MathSciNet  Google Scholar 

  3. Bressan A, Yang T. On the convergence rate of vanishing viscosity approximations. Comm Pure Appl Math, 2004, 57: 1075–1109

    Article  MATH  MathSciNet  Google Scholar 

  4. DiPerna R J, Lions P L. Ordinary differential equations, transport theory, and Sobolev spaces. Invent Math, 1989, 98: 511–547

    Article  MATH  MathSciNet  Google Scholar 

  5. Gimse T. Conservation laws with discontinuous flux functions. SIAM J Math Anal, 1993, 24: 279–289

    Article  MATH  MathSciNet  Google Scholar 

  6. Gosse L, James F. Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients. Math Comp, 2000, 69: 987–1015

    Article  MATH  MathSciNet  Google Scholar 

  7. Jin S, Markowich P A, Sparber C. Mathematical and computational methods for semiclassical Schrodinger equations. Acta Numer, 2011, 20: 211–289

    Article  MathSciNet  Google Scholar 

  8. Jin S, Novak K. A semiclassical transport model for two-dimensional thin quantum barriers. J Comput Phys, 2007, 226: 1623–1644

    Article  MATH  MathSciNet  Google Scholar 

  9. Jin S, Wen X. Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials. Commun Math Sci, 2005, 3: 285–315

    Article  MATH  MathSciNet  Google Scholar 

  10. Jin S, Wen X. A Hamiltonian-preserving scheme for the Liouville equation of geometrical optics with partial transmissions and reflections. SIAM J Numer Anal, 2006, 44: 1801–1828

    Article  MATH  MathSciNet  Google Scholar 

  11. Karlsen K H, Klingenberg C, Risebro N H. A relaxation scheme for conservation laws with a discontinuous coefficient. Math Comp, 2004, 73: 1235–1259

    Article  MATH  MathSciNet  Google Scholar 

  12. Karlsen K H, Towers J D. Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux. Chin Ann Math Ser B, 2004, 25: 287–318

    Article  MATH  MathSciNet  Google Scholar 

  13. Klingenberg C, Risebro N H. Convex conservation laws with discontinuous coefficients: Existence, uniqueness and asymptotic behavior. Comm Partial Differential Equations, 1995, 20: 1959–1990

    Article  MATH  MathSciNet  Google Scholar 

  14. Kuznetsov N N. Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation. USSR Comp Math Math Phys, 1976, 16: 105–119

    Article  Google Scholar 

  15. LeVeque R J, Li Z L. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J Numer Anal, 1994, 31: 1019–1044

    Article  MATH  MathSciNet  Google Scholar 

  16. Lin L W, Temple B J, Wang J H. A comparison of convergence rates for Godunov’s method and Glimm’s method in resonant nonlinear systems of conservation laws. SIAM J Numer Anal, 1995, 32: 824–840

    Article  MATH  MathSciNet  Google Scholar 

  17. Mayo A. The fast solution of Poisson’s and the biharmonic equations on irregular regions. SIAM J Sci Comp, 1984, 21: 285–299

    MATH  MathSciNet  Google Scholar 

  18. Mishra S. Convergence of upwind finite difference schemes for a scalar conservation law with indefinite discontinuities in the flux function. SIAM J Numer Anal, 2005, 43: 559–577

    Article  MATH  MathSciNet  Google Scholar 

  19. Poupaud F, Rascle M. Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients. Comm Partial Differential Equations, 1997, 22: 337–358

    Article  MATH  MathSciNet  Google Scholar 

  20. Peskin C S. Numerical analysis of blood flow in the heart. J Comput Phys, 1977, 25: 220–252

    Article  MATH  MathSciNet  Google Scholar 

  21. Sabac F. The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws. SIAM J Numer Anal, 1997, 34: 2306–2318

    Article  MATH  MathSciNet  Google Scholar 

  22. Sanders R. On convergence of monotone finite difference schemes with variable spatial differencing. Math Comp, 1983, 40: 91–106

    Article  MATH  MathSciNet  Google Scholar 

  23. Tang T, Teng Z H. The sharpness of Kuznetsov’s \(O\left( {\sqrt {\Delta x} } \right)\) L 1-error estimate for monotone difference schemes. Math Comp, 1995, 64: 581–589

    MATH  MathSciNet  Google Scholar 

  24. Tang T, Teng Z H. Viscosity methods for piecewise smooth solutions to scalar conservation laws. Math Comp, 1997, 66: 495–526

    Article  MATH  MathSciNet  Google Scholar 

  25. Tang T, Teng Z H, Xin Z P. Fractional rate of convergence for viscous approximation to nonconvex conservation laws. SIAM J Math Anal, 2003, 35: 98–122

    Article  MATH  MathSciNet  Google Scholar 

  26. Teng Z H, Zhang P W. Optimal L 1-rate of convergence for the viscosity method and monotone scheme to piecewise constant solutions with shocks. SIAM J Numer Anal, 1997, 34: 959–978

    Article  MATH  MathSciNet  Google Scholar 

  27. Towers J D. Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J Numer Anal, 2000, 38: 681–698

    Article  MATH  MathSciNet  Google Scholar 

  28. Wen X. Convergence of an immersed interface upwind scheme for linear advection equations with piecewise constant coefficients II: Some related binomial coefficients inequalities. J Comput Math, 2009, 27: 474–483

    Article  MATH  MathSciNet  Google Scholar 

  29. Wen X, Jin S. Covergence of an immersed interface upwind scheme for linear advection equations with piecewise constant coefficients I: L 1-error estimates. J Comput Math, 2008, 26: 1–22

    MATH  MathSciNet  Google Scholar 

  30. Wen X, Jin S. The l 1-error estimates for a Hamiltonian-preserving scheme for the Liouville equation with piecewise constant potentials. SIAM J Num Anal, 2008, 46: 2688–2714

    Article  MATH  MathSciNet  Google Scholar 

  31. Zhang C, LeVeque R J. The immersed interface method for acoustic wave equations with discontinuous coefficients. Wave Motion, 1997, 25: 237–263

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Institute of Natural Sciences and MOE Key Lab in Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai, 200240, China

    Shi Jin

  2. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Shi Jin & Peng Qi

Authors
  1. Shi Jin
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  2. Peng Qi
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Correspondence to Shi Jin.

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Dedicated to Professor Shi Zhong-Ci on the Occasion of his 80th Birthday

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Jin, S., Qi, P. l 1-error estimates on the immersed interface upwind scheme for linear convection equations with piecewise constant coefficients: A simple proof. Sci. China Math. 56, 2773–2782 (2013). https://doi.org/10.1007/s11425-013-4738-2

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  • DOI: https://doi.org/10.1007/s11425-013-4738-2

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