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Space–Time Adaptive Solution of First Order PDES

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Abstract

An explicit time-stepping method is developed for adaptive solution of time-dependent partial differential equations with first order derivatives. The space is partitioned into blocks and the grid is refined and coarsened in these blocks. The equations are integrated in time by a Runge–Kutta–Fehlberg (RKF) method. The local errors in space and time are estimated and the time and space steps are determined by these estimates. The method is shown to be stable if one-sided space discretizations are used. Examples such as the wave equation, Burgers’ equation, and the Euler equations in one space dimension with discontinuous solutions illustrate the method.

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References

  1. G. Beckett J.A. Mackenzie A. Ramage D.M. Sloan (2001) ArticleTitleOn the numerical solution of one-dimensional PDEs using adaptive methods based on equidistribution J. Comput. Phys. 167 372–392 Occurrence Handle10.1006/jcph.2000.6679 Occurrence Handle0985.65097

    Article  MATH  Google Scholar 

  2. M. Berger P. Colella (1989) ArticleTitleLocal adaptive mesh refinement for shock hydrodynamics J. Comput. Phys. 82 64–84 Occurrence Handle10.1016/0021-9991(89)90035-1 Occurrence Handle0665.76070

    Article  MATH  Google Scholar 

  3. M. Berger R. LeVeque (1998) ArticleTitleAdaptive mesh refinement using wave-propagation algorithms for hyperbolic systems SIAM J. Numer. Anal. 35 2298–2316 Occurrence Handle10.1137/S0036142997315974 Occurrence Handle1655847 Occurrence Handle0921.65070

    Article  MathSciNet  MATH  Google Scholar 

  4. W. Cao W. Huang R.D. Russell (1999) ArticleTitleAn r-adaptive finite element method based upon moving mesh PDEs J. Comput. Phys. 149 221–244 Occurrence Handle10.1006/jcph.1998.6151 Occurrence Handle1672735 Occurrence Handle0923.65062

    Article  MathSciNet  MATH  Google Scholar 

  5. B. Engquist S. Osher (1980) ArticleTitleStable and entropy satisfying approximations for transonic flow calculations Math. Comp. 34 45–75 Occurrence Handle551290 Occurrence Handle0438.76051

    MathSciNet  MATH  Google Scholar 

  6. B. Engquist B. Sjögreen (1998) ArticleTitleThe convergence of finite difference schemes in the presence of shocks SIAM J. Numer. Anal. 35 2464–2485 Occurrence Handle10.1137/S0036142997317584 Occurrence Handle1655855 Occurrence Handle0922.76254

    Article  MathSciNet  MATH  Google Scholar 

  7. Ferm, L., and Lötstedt, P. (2001). Efficiency in the adaptive solution of inviscid compressible flow problems, in Proceedings of WCNA 2000. Nonlinear Anal. 47, 3467–3478

  8. L. Ferm P. Lötstedt (2002) ArticleTitleAdaptive error control for steady state solutions of inviscid flow SIAM J. Sci. Comput. 23 1777–1798 Occurrence Handle10.1137/S1064827500367452 Occurrence Handle1885083 Occurrence Handle1008.76051

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Ferm P. Lötstedt (2003) ArticleTitleAnisotropic grid adaptation for Navier-Stokes’ equations J. Comput. Phys. 190 22–41 Occurrence Handle10.1016/S0021-9991(03)00250-X Occurrence Handle2046757 Occurrence Handle1027.76030

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Ferm P. Lötstedt (2004) ArticleTitleAccurate and stable grid interfaces for finite volume methods Appl. Numer. Math. 49 207–224 Occurrence Handle10.1016/j.apnum.2003.07.004 Occurrence Handle2045499 Occurrence Handle1055.65103

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Gottlieb C.-W. Shu (1998) ArticleTitleTotal variation diminishing Runge-Kutta schemes Math. Comp. 67 73–85 Occurrence Handle10.1090/S0025-5718-98-00913-2 Occurrence Handle1443118 Occurrence Handle0897.65058

    Article  MathSciNet  MATH  Google Scholar 

  12. K. Gustafsson (1991) ArticleTitleControl theoretic techniques for stepsize selection in explicit Runge-Kutta methods ACM Trans. Math. Software 17 533–554 Occurrence Handle10.1145/210232.210242 Occurrence Handle0900.65256 Occurrence Handle1140040

    Article  MATH  MathSciNet  Google Scholar 

  13. E. Hairer S.P. Nørsett G. Wanner (1993) Solving Ordinary Differential Equations EditionNumber2 Springer-Verlag Berlin Occurrence Handle0789.65048

    MATH  Google Scholar 

  14. K. Hörnell (1999) Runge-Kutta Time Step Selection for Flow Problems Uppsala Dissertations 16, Faculty of Science and Technology, Uppsala University Uppsala, Sweden

    Google Scholar 

  15. K. Hörnell P. Lötstedt (2001) ArticleTitleTime step selection for shock problems Commun. Numer. Meth. Engng 17 477–484 Occurrence Handle10.1002/cnm.423 Occurrence Handle0985.65115

    Article  MATH  Google Scholar 

  16. R.D. Hornung J.A. Trangenstein (1997) ArticleTitleAdaptive mesh refinement and multilevel iteration for flow in porous media J. Comput. Phys. 136 522–545 Occurrence Handle10.1006/jcph.1997.5779 Occurrence Handle1474414 Occurrence Handle0898.76074

    Article  MathSciNet  MATH  Google Scholar 

  17. W. Huang R.D. Russell (1999) ArticleTitleMoving mesh strategy based on a gradient flow equation for two-dimensional problems SIAM J. Sci. Comput. 20 998–1015 Occurrence Handle1665654 Occurrence Handle0956.76076

    MathSciNet  MATH  Google Scholar 

  18. J.P. Jessee W.A. Fiveland L.H. Howell P. Colella R.B. Pember (1998) ArticleTitleAn adaptive mesh refinement algorithm for the radiative transport equation J. Comput. Phys. 139 380–398 Occurrence Handle10.1006/jcph.1997.5870 Occurrence Handle0905.65131

    Article  MATH  Google Scholar 

  19. C. Johnson A. Szepessy (1995) ArticleTitleAdaptive finite element methods for conservation laws based on a posteriori error estimates Comm. Pure Appl. Math. 48 199–234 Occurrence Handle1322810 Occurrence Handle0826.65088

    MathSciNet  MATH  Google Scholar 

  20. S. Karni A. Kurganov G. Petrova (2002) ArticleTitleA smoothness indicator for adaptive algorithms for hyperbolic systems J. Comput. Phys. 178 323–341 Occurrence Handle10.1006/jcph.2002.7024 Occurrence Handle1899180 Occurrence Handle0998.65092

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Keppens M. Nool G. Tóth J.P. Goedbloed (2003) ArticleTitleAdaptive mesh refinement for conservative systems: multi-dimensional efficiency evaluation Comput. Phys. Comm. 153 317–339 Occurrence Handle10.1016/S0010-4655(03)00139-5 Occurrence Handle2009189

    Article  MathSciNet  Google Scholar 

  22. H.-O. Kreiss J. Lorenz (1989) Initial Boundary Value Problems and the Navier-Stokes Equations Academic Press Boston Occurrence Handle0689.35001

    MATH  Google Scholar 

  23. B. Leer Particlevan (1984) ArticleTitleOn the relation between the upwind-differencing schemes of Godunov, Engquist-Osher, and Roe SIAM J. Sci. Comput. 5 1–20 Occurrence Handle0547.65065

    MATH  Google Scholar 

  24. R.J. LeVeque (2002) Finite Volume Methods for Hyperbolic Problems Cambridge University Press Cambridge Occurrence Handle1010.65040

    MATH  Google Scholar 

  25. R. Li W.B. Liu H.P. Ma (2004) ArticleTitleMoving mesh method with error-estimator-based monitor and its applications to static obstacle problem J. Sci. Comput. 21 31–55 Occurrence Handle10.1023/B:JOMP.0000027954.83289.00 Occurrence Handle2064325

    Article  MathSciNet  Google Scholar 

  26. R. Li T. Tang P. Zhang (2001) ArticleTitleMoving mesh methods in multiple dimensions based on harmonic maps J. Comput. Phys. 170 562–588 Occurrence Handle10.1006/jcph.2001.6749 Occurrence Handle1844903 Occurrence Handle0986.65090

    Article  MathSciNet  MATH  Google Scholar 

  27. F. Liu S. Ji G. Liao (1998) ArticleTitleAn adaptive grid method and its application to steady Euler flow calculations SIAM J. Sci. Comput. 20 811–825 Occurrence Handle10.1137/S1064827596305738 Occurrence Handle1648876

    Article  MathSciNet  Google Scholar 

  28. P. Lötstedt A. SöderbergS. Ramage L. Hemmingsson-Frändén (2002) ArticleTitleImplicit solution of hyperbolic equations with space-time adaptivity BIT 42 134–158 Occurrence Handle10.1023/A:1021978304268 Occurrence Handle1896390 Occurrence Handle0999.65084

    Article  MathSciNet  MATH  Google Scholar 

  29. S. Osher F. Solomon (1982) ArticleTitleUpwind difference schemes for hyperbolic systems of conservation laws Math. Comp. 38 339–374 Occurrence Handle645656 Occurrence Handle0483.65055

    MathSciNet  MATH  Google Scholar 

  30. K.G. Powell (1994) ArticleTitleA tree-based adaptive scheme for solution of the equations of gas dynamics and magnetohydrodynamics Appl. Numer. Math. 14 327–352 Occurrence Handle10.1016/0168-9274(94)90032-9 Occurrence Handle0803.76061 Occurrence Handle1273831

    Article  MATH  MathSciNet  Google Scholar 

  31. G. Sod (1978) ArticleTitleA survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws J. Comput. Phys. 27 32–78 Occurrence Handle10.1016/0021-9991(78)90023-2 Occurrence Handle495002

    Article  MathSciNet  Google Scholar 

  32. G. Söderlind (2002) ArticleTitleAutomatic control and adaptive time-stepping Numer. Alg. 31 281–310 Occurrence Handle1012.65080

    MATH  Google Scholar 

  33. J.M. Stockie J.A. Mackenzie R.D. Russell (2001) ArticleTitleA moving mesh method for one-dimensional hyperbolic conservations laws SIAM J. Sci. Comput. 22 1791–1813 Occurrence Handle1813298 Occurrence Handle10.1137/S1064827599364428 Occurrence Handle0989.65096

    Article  MathSciNet  MATH  Google Scholar 

  34. H. Tang T. Tang (2003) ArticleTitleAdaptive mesh methods for one- and two-dimensional hyperbolic conservations laws SIAM J. Numer. Anal. 41 487–515 Occurrence Handle10.1137/S003614290138437X Occurrence Handle2004185 Occurrence Handle1052.65079

    Article  MathSciNet  MATH  Google Scholar 

  35. T. Tang Z.-H. Teng (1997) ArticleTitleViscosity methods for piecewise smooth solutions to scalar conservation laws Math. Comp. 66 495–526 Occurrence Handle10.1090/S0025-5718-97-00822-3 Occurrence Handle1397446 Occurrence Handle0864.65060

    Article  MathSciNet  MATH  Google Scholar 

  36. A. Vande Wouwer P. Saucez W.E. Schiesser (Eds) (2001) Adaptive Method of Lines Chapman & Hall Boca Raton Occurrence Handle0986.65083

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Information Technology, Division of Scientific Computing, Uppsala University, SE-75105, Uppsala, Sweden

    Lars Ferm & Per Lötstedt

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  1. Lars Ferm
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  2. Per Lötstedt
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Correspondence to Lars Ferm.

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Ferm, L., Lötstedt, P. Space–Time Adaptive Solution of First Order PDES. J Sci Comput 26, 83–110 (2006). https://doi.org/10.1007/s10915-004-4801-9

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  • DOI: https://doi.org/10.1007/s10915-004-4801-9

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