Daniel Appelö: What’s new with the wave equation?

Abstract:

The defining feature of waves is their ability to propagate over vast distances in space and time without changing shape. This unique property enables the transfer of information and constitutes the foundation of today’s communication based society. To see that accurate propagation of waves requires high order accurate numerical methods, consider the problem of propagating a wave in three dimensions for 100 wavelengths with 1% error. Using a second order method this requires 0.2 trillion space-time samples while a sixth order method requires 87,000!

In this talk we present new arbitrary order dissipative and conservative Hermite methods for the scalar wave equation. The degrees-of-freedom of Hermite methods are tensor-product Taylor polynomials of degree m in each coordinate centered at the nodes of Cartesian grids, staggered in time. The methods achieve space-time accuracy of order O(2m). Besides their high order of accuracy in both space and time combined, they have the special feature that they are stable for CFL = 1, for all orders of accuracy. This is significantly better than standard high-order element methods. Moreover, the large time steps are purely local to each cell, minimizing communication and storage requirements.

Bio:

Daniel Appelö holds a Ph. D. degree in Numerical Analysis from the Royal Institute of Technology in Sweden and is currently an Associate Professor in Applied Mathematics and a BOLD (Broadening Opportunity through Leadership and Diversity) Faculty Fellow at University of Colorado, Boulder. Prior to joining CU he was an Associate Professor at University of New Mexico and held postdoctoral positions at California Institute of Technology and Lawrence Livermore National Laboratory.