### 2006-20

by Heineken, W., Kunik, M..

**Series:** 2006-20, Preprints

- MSC:
- 35-04 Explicit machine computation and programs (not the theory of computation or programming)
- 35L60 Nonlinear first-order hyperbolic equations
- 35L65 Conservation laws
- 35L67 Shocks and singularities
- 35Q05 ~Euler-Poisson-Darboux equations
- 35Q75 PDEs in connection with relativity and gravitational theory

**Abstract:**

A finite volume method with grid adaption is applied

to two hyperbolic problems: the ultra-relativistic Euler equations,

and a scalar conservation law. Both problems are considered

in two space dimensions and share the common feature

of moving shock waves. In contrast to the classical Euler equations,

the derivation of appropriate initial conditions

for the ultra-relativistic Euler equations is a non-trivial problem

that is solved using one-dimensional shock conditions

and the Lorentz invariance of the system.

The discretization of both problems is based on a finite volume method

of second order in both space and time on a triangular grid.

We introduce a variant of the min-mod limiter that avoids unphysical

states for the Euler system. The grid is adapted during the integration

process. The frequency of grid adaption is controlled automatically

in order to guarantee a fine resolution of the moving shock fronts.

We introduce the concept of ``width refinement'' which enlarges the

width of strongly refined regions around the shock fronts;

the optimal width is found by a numerical study.

As a result we are able to improve efficiency by decreasing

the number of adaption steps. The performance of the finite volume

scheme is compared with several lower order methods.

**Keywords:**

Scalar conservation laws, ultra-relativistic Euler equations, Riemann problem, shock waves, Lorentz-transformations, finite volume method, min-mod limiter, grid adaption