The Optimal Convergence Rate of Monotone Schemes for Conservation Laws in the Wasserstein Distance

In 1994, Nessyahu, Tadmor and Tassa studied convergence rates of monotone finite volume approximations of conservation laws. For compactly supported, $\Lip^+$-bounded initial data they showed a first-order convergence rate in the Wasserstein distance. Our main result is to prove that this rate is optimal. We further provide numerical evidence indicating that the rate in the case of $\Lip^+$-unbounded initial data is worse than first-order.

[1]  Wei-Cheng Wang,et al.  On L 1 convergence rate of viscous and numerical approximate solutions of genuinely nonlinear scalar conservation laws , 1999 .

[2]  Eduard Harabetian,et al.  Rarefactions and large time behavior for parabolic equations and monotone schemes , 1988 .

[3]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[4]  Tao Tang,et al.  The sharpness of Kuznetsov’s $O(\sqrt{\Delta x})\ L\sp 1$-error estimate for monotone difference schemes , 1995 .

[5]  C. Villani.  Topics in Optimal Transportation , 2003 .

[6]  Ulrik S. Fjordholm,et al.  Second-Order Convergence of Monotone Schemes for Conservation Laws , 2016, SIAM J. Numer. Anal..

[7]  N. S. Bakhvalov,et al.  Estimation of the error of numerical integration of a first-order quasilinear equation , 1962 .

[8]  Tao Tang,et al.  The sharpness of Kuznetsov's O D x L 1 -error estimate for monotone difference schemes , 1995 .

[9]  Susanne Solem,et al.  Convergence Rates of the Front Tracking Method for Conservation Laws in the Wasserstein Distances , 2018, SIAM J. Numer. Anal..

[10]  N. SIAMJ.,et al.  THE OPTIMAL CONVERGENCE RATE OF MONOTONE FINITE DIFFERENCE METHODS FOR HYPERBOLIC CONSERVATION LAWS∗ , 1997 .

[11]  E. Tadmor.  Local error estimates for discontinuous solutions of nonlinear hyperbolic equations , 1991 .

[12]  Tamir Tassa,et al.  The convergence rate of Godunov type schemes , 1994 .

[13]  Eitan Tadmor,et al.  The convergence rate of approximate solutions for nonlinear scalar conservation laws. Final Report , 1991 .

[14]  Tamir Tassa,et al.  Convergence rate of approximate solutions to conservation laws with initial rarefactions , 1994 .

[15]  James M. Hyman,et al.  On Finite-Difference Approximations and Entropy Conditions for Shocks , 2015 .

[16]  Pingwen Zhang,et al.  Optimal L 1 -Rate of Convergence for The Viscosity Method and Monotone Scheme to Piecewise Constant Solutions with Shocks , 1997 .

[17]  N. N. Kuznetsov.  Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation , 1976 .