SCS: Jeff Banks (RPI)

Tuesday Sep 30, 2025, 10:00 AM → 11:00 AM America/New_York

Title: High-order accurate schemes for time and frequency domain wave equations

For many engineering problems, e.g. nanophotonic devices, high-order accurate numerical methods are desirable because they are potentially orders of magnitude more efficient than their low-order counterparts. However, realizing the potential payo of high-order methods in complex domains with multiple materials, particularly for problems involving wave propagation, has proven challenging. In this talk I discuss our recent work on high-order accurate methods for wave systems. In part I, the focus will be time-domain simulation of dispersive and nondispersive Maxwell's equations. Complex geometry is treated with overset grids, and interfaces between different materials are accurately and e ciently treated using compatibility coupling conditions. Compatibility coupling is a key ingredient to our approach, and when combined with high-order accurate modi ed equation time stepping yields a scheme with large CFL-one time step restriction. In part II of the talk, I will discuss the use of these time-domain solvers to obtain the solution to Helmholtz equations. The solvers are based on the WaveHoltz algorithm which computes solutions of the Helmholtz equation by time- ltering solutions of a corresponding wave equation. This approach avoids the need to invert an inde nite matrix, which can cause convergence di culties for many iterative solvers for Helmholtz problems. The solution of the wave equation can be solved e ciently with implicit time-stepping using as few as ve time-steps per period, independent of the mesh size. When multigrid is used to solve the implicit time-stepping equations, the cost of the resulting WaveHoltz scheme scales linearly with the total number of grid points N, at fixed frequency.