**4 - Adaptive Refinement**

The accurate numerical solution of partial differential equations requires large computational resources. The execution times are long and large memories are needed to store the solution. This is especially so for the equations of compressible fluid flow. There are at least three ways of improving the situation:

- Use a parallel computer with a large internal memory.
- Adapt the grid to the solution so that the cells are concentrated in areas where they reduce the numerical error and avoid waste of cells.
- Speed up the calculations by an efficient solver of the nonlinear equations of the steady state solution or in each time-step of an implicit solver of the time-dependent equations.

Here, we concentrate on the econd issue, in particular on the adaption of structured grids. Structured grids usually require less memory than unstructured grids and the solvers on them are often faster. They are suitable for processors with limited cache memories. Furthermore, there are many well validated programs used for production calculations in the industry based on structured grids where better efficiency is needed.

AMRCLAW combines the CLAWPACK routines with the Marsha Berger's adaptive mesh refinement ( AMR ) algorithm, which was originally developed for gas dynamics. Once a Riemann solver and boundary conditions have been implemented in CLAWPACK to solve a particular problem, it is quite easy to swith to AMRCLAW and use adaptively refined grids, with the potential for large savings in CPU time for multi-dimentional problems. Berger's original AMR code applies on Cartesian grids; By merging it with CLAWPACK it is now straightforward to apply it on logically-rectangular curvilinear grids as well.