# Wavelet Adaptive Multi-Resolution

## Wavelet Adaptive Multi-Resolution

Adaptive grids are important in many computational problems. Wavelet Adaptive Multi-Resolution (WAMR) uses the expansion of functions in a wavelet basis to generate the unstructured computational grid. Dendro-GR uses WAMR in simulations of binary black holes.

### WAMR and Relativistic Fluids

The Riemann problem in fluid dynamics is a solution for the evolution of two discontinuous states. The solution has two waves, which can be either shocks or rarefaction waves. The plots below show a solution with a rarefaction wave (moving left) and a shock (moving right) at a fixed time. The top row shows the fluid density (ρ) and velocity (*v*), and the bottom left frame shows the pressure (*P*). The bottom right frame shows all of the points used to calculate the solution, with the vertical axis showing the different grid levels. The wavelet algorithm refines on non-smooth features of the solution, such as the shock and contact discontinuities, as well as the left and right end-points of the rarefaction wave.

This figure shows the points used to calculate a Riemann problem in two dimensions with a bow shock.

### The Non-Linear Sigma Model

A simple example for Dendro and WAMR is a wave equation with a non-linear potential, such as the non-linear sigma model [1]. This movie shows an evolution of two interacting pulses.

### References:

1. Steven L. Liebling, "Singularity threshold of the nonlinear sigma model using 3D adaptive mesh refinement," *Phys. Rev.* D **66**, 041703 (2002). DOI