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Relaxation Approximations and Relaxation Schemes for PDEs
Probability| Speaker: | Shi Jin, Mathematics, Georgia Institute of Technology |
| Location: | 693 Kerr |
| Start time: | Tue, Mar 16 1999, 4:10PM |
Description
The classical solutions to many partial differential equations
fail to exist in finite time even if the initial data are smooth.
Classical examples include hyperbolic conservation laws and
Hamilton-Jacobi equations. In such cases viscosity solutions have
been introduced which allow the selection of physically relevant
weak solutions beyond the singularity time.
We introduce the relaxation approximation to such partial differential
equations by replacing such equations with a semi-linear hyperbolic
systems with stiff relaxations. While the viscosity regularization
arises from the Navier-Stokes approximation to the Euler equations,
the relaxation approximation is analogous to the regularization of
the Euler equations by the more fundamental Boltzmann equation.
The relaxation approximation introduces a physically natural
way to select the entropy or viscosity solution to hyperbolic conservation
laws and Hamilton-Jacobi equation. The semilinear nature of
the relaxation system paves a new way to derive
numerical schemes ( known as the relaxation schemes) that
are simple, efficient, and Riemann solver free, with a
high resolution, for problems involving shocks, fronts, or other type of
discontinuous solutions.