# zbMATH — the first resource for mathematics

Local error analysis for approximate solutions of hyperbolic conservation laws. (English) Zbl 1127.65070
Summary: We consider approximate solutions to nonlinear hyperbolic conservation laws. If the exact solution is unavailable, the truncation error may be the only quantitative measure for the quality of the approximation. We propose a new way of estimating the local truncation error, through the use of localized test-functions. In the convex scalar case, they can be converted into $$L_{\text{loc}}^{\infty}$$ estimates following the $$Lip'$$ convergence theory developed by E. Tadmor and T. Tang [SIAM J. Numer. Anal. 36, No. 6, 1739–1758 (1999; Zbl 0934.35088)]. Comparisons between the local truncation error and the $$L_{\text{loc}}^{\infty}$$-error show remarkably similar behavior.
Numerical results are presented for the convex scalar case, where the theory is valid, as well as for nonconvex scalar examples and the Euler equations of gas dynamics. The local truncation error has proved a reliable smoothness indicator and has been implemented in adaptive algorithms by the authors and G. Petrova [J. Comput. Phys. 178, 323–341 (2002; Zbl 0998.65092)].

##### MSC:
 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws
Full Text:
##### References:
 [1] B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor,Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, C.I.M.E. Course in Cetraro, Italy, June 1997, ed. A. Quarteroni, Lecture Notes in Mathematics, Vol. 1697 (Springer, New York, 1998). · Zbl 0904.00047 [2] C. de Boor,A Practical Guide to Splines, Applied Mathematical Sciences, Vol. 27 (Springer, New York, 1978). [3] E. Godlewski and P.-A. Raviart,Numerical Approximation of Hyperbolic Systems of Conservation Laws (Springer, New York, 1996). [4] P. Houston and E. Süli, Local a posteriori error indicators for hyperbolic problems, Technical Report NA-97/14, Oxford University Computing Laboratory (1997), http://web.comlab.ox.ac.uk/oucl/publications/natr/na-97-14.html. [5] Karni, S.; Kurganov, A.; Petrova, G., A smoothness indicator for adaptive algorithms for hyperbolic systems, J. Comput. Phys., 178, 323-341, (2002) · Zbl 0998.65092 [6] D. Kröner,Numerical Schemes for Conservation Laws (Wiley, Chichester, 1997). · Zbl 0872.76001 [7] A. Kurganov, Conservation laws: stability of numerical approximations and nonlinear regularization, Ph.D. thesis, Tel-Aviv University, Israel (1997). [8] Kurganov, A.; Noelle, S.; Petrova, G., Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., 23, 707-740, (2001) · Zbl 0998.65091 [9] R. LeVeque,Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics (Cambridge Univ. Press, Cambridge, 2002). · Zbl 1010.65040 [10] Nessyahu, H.; Tadmor, E., The convergence rate of approximate solutions for nonlinear scalar conservation laws, SIAM J. Numer. Anal., 29, 1505-1519, (1992) · Zbl 0765.65092 [11] Nessyahu, H.; Tadmor, E.; Tassa, T., The convergence rate of Godunov type schemes, SIAM J. Numer. Anal., 31, 1-16, (1994) · Zbl 0799.65096 [12] R. Richtmyer and K.W. Morton,Difference Methods for Initial-Value Problems, 2nd ed. (Interscience, New York, 1967). · Zbl 0155.47502 [13] Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43, 357-372, (1981) · Zbl 0474.65066 [14] J. Smoller,Shock Waves and Reaction Diffusion-Equations, 2nd ed., Grundleheren Series, Vol. 258 (Springer, New York, 1994). [15] Sod, G., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 22, 1-31, (1978) · Zbl 0387.76063 [16] E. Süli, A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems, Lecture Notes in Computer Science and Engineering, eds. D. Kröner, M. Ohlberger and C. Rhode, Vol. 5 (Springer, New York, 1998); also, Technical Report NA-97/21, Oxford University Computing Laboratory (1997); http://web.comlab.ox.ac.uk/oucl/publications/natr/na-97-21.html. [17] Tadmor, E., Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal., 28, 891-906, (1991) · Zbl 0732.65084 [18] Tadmor, E.; Tang, T.; Fey, M. (ed.); Jeltsch, R. (ed.), Pointwise convergence rate for nonlinear conservation laws, 925-934, (1999), Basel · Zbl 0928.35100 [19] Tadmor, E.; Tang, T., Pointwise error estimates for scalar conservation laws with piecewise smooth solutions, SIAM J. Numer. Anal., 36, 1739-1756, (1999) · Zbl 0934.35088
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.