# Computational Mathematics and Scientific Computing Seminar

## High Order Accurate Compact Finite Difference Scheme for Wave Equation in General Shaped Domains using Simple Meshes

**Speaker:**
Eli Turkel, Tel Aviv University

**Location:**
Warren Weaver Hall 1302

**Date:**
Nov. 2, 2018, 10 a.m.

**Synopsis:**

We consider the wave equation in a general shaped domain perhaps including general interfaces. We consider two fourth order accurate algorithms that use only simple (Cartesian or Polar) grids. Both approaches rely on the difference potential approach.

In one approach we discretize implicitly in time with fourth order accuracy which leads to a modified Helmholtz equation. (2D or 3D with a variable speed of sound) for the upper time level. This modified Helmholtz equation is then approximated by a fourth order, in space, scheme. Difference potentials are then used to solve for the general shaped domain using Cartesian grids.

The other approach employs the difference potentials directly in 3D+time and again the shape of the boundary does not have to conform to the grid, on which the difference potentials are computed. A fourth order discretization is used in both time and space. Furthermore, it takes advantage of Huygens' principle. F\So for long simulation times, the method offers sub-linear complexity, i.e., appears asymptotically cheaper than a typical explicit scheme.

Results are presented for both variations.