A posteriori Error Estimates for Numerical Solutions to Hyperbolic Conservation Laws

A posteriori Error Estimates for Numerical Solutions to Hyperbolic Conservation Laws

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Maria Teresa Chiri, Penn State University

Zoom link: https://princeton.zoom.us/j/4745473988

For general n x n hyperbolic systems of conservation laws in one space dimension, it is well known that the Cauchy problem has a unique entropy-weak solution, depending continuously on the initial data.  Assuming small total variation, a priori estimates on the L^1 distance between an approximate solution and the exact solution have been obtained in connection with (i) front tracking approximations, (ii) the Glimm scheme, and (iii) vanishing viscosity approximations. However, no a priori estimate is yet known for approximate solutions obtained by fully discrete schemes, such as the Lax-Friedrichs or the Godunov scheme. Taking a different point of view, we will focus on a posteriori error estimates. These are achieved by a "post-processing algorithm" that checks the total variation of the numerically computed solution, and computes its oscillation on suitable subdomains.