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An Optimal O(N) Helmholtz Solver for Complex Geometry using WaveHoltz and Overset Grids

Daniel Appelo, Jeffrey W. Banks, William D. Henshaw, Donald W. Schwendeman

TL;DR

This work presents an efficient, high-order Helmholtz solver for complex geometries by combining the WaveHoltz time-filter approach with overset grids and high-order discretizations. The core idea is to solve a time-domain wave equation with implicit time-stepping and a fixed number of steps per period, then extract the Helmholtz solution via a time-average filter, avoiding indefinite-system inversions. Acceleration is achieved via GMRES and deflation of slow eigenmodes, with multigrid enabling linear-scaling performance in $N$ at a fixed frequency. The authors derive a concrete pollution-error framework and a practical PPW rule of thumb to guide discretization, and validate the method across 2D and 3D geometries, showing robust convergence and near-optimal $O(N)$ scaling in realistic settings.

Abstract

We develop efficient and high-order accurate solvers for the Helmholtz equation on complex geometry. The schemes are based on the WaveHoltz algorithm which computes solutions of the Helmholtz equation by time-filtering solutions of the wave equation. The approach avoids the need to invert an indefinite matrix which can cause convergence difficulties for many iterative solvers for indefinite Helmholtz problems. Complex geometry is treated with overset grids which use Cartesian grids throughout most of the domain together with curvilinear grids near boundaries. The basic WaveHoltz fixed-point iteration is accelerated using GMRES and also by a deflation technique using a set of precomputed eigenmodes. The solution of the wave equation is solved efficiently with implicit time-stepping using as few as five time-steps per period, independent of the mesh size. The time-domain solver is adjusted to remove dispersion errors in time and this enables the use of such large time-steps without degrading the accuracy. 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) and is thus optimal in CPU-time and memory usage as the mesh is refined. A simple rule-of-thumb formula is provided to estimate the number of points-per-wavelength required for a p-th order accurate scheme which accounts for pollution (dispersion) errors. Numerical results are given for problems in two and three space dimensions, to second and fourth-order accuracy, and they show the potential of the approach to solve a wide range of large-scale problems.

An Optimal O(N) Helmholtz Solver for Complex Geometry using WaveHoltz and Overset Grids

TL;DR

This work presents an efficient, high-order Helmholtz solver for complex geometries by combining the WaveHoltz time-filter approach with overset grids and high-order discretizations. The core idea is to solve a time-domain wave equation with implicit time-stepping and a fixed number of steps per period, then extract the Helmholtz solution via a time-average filter, avoiding indefinite-system inversions. Acceleration is achieved via GMRES and deflation of slow eigenmodes, with multigrid enabling linear-scaling performance in at a fixed frequency. The authors derive a concrete pollution-error framework and a practical PPW rule of thumb to guide discretization, and validate the method across 2D and 3D geometries, showing robust convergence and near-optimal scaling in realistic settings.

Abstract

We develop efficient and high-order accurate solvers for the Helmholtz equation on complex geometry. The schemes are based on the WaveHoltz algorithm which computes solutions of the Helmholtz equation by time-filtering solutions of the wave equation. The approach avoids the need to invert an indefinite matrix which can cause convergence difficulties for many iterative solvers for indefinite Helmholtz problems. Complex geometry is treated with overset grids which use Cartesian grids throughout most of the domain together with curvilinear grids near boundaries. The basic WaveHoltz fixed-point iteration is accelerated using GMRES and also by a deflation technique using a set of precomputed eigenmodes. The solution of the wave equation is solved efficiently with implicit time-stepping using as few as five time-steps per period, independent of the mesh size. The time-domain solver is adjusted to remove dispersion errors in time and this enables the use of such large time-steps without degrading the accuracy. 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) and is thus optimal in CPU-time and memory usage as the mesh is refined. A simple rule-of-thumb formula is provided to estimate the number of points-per-wavelength required for a p-th order accurate scheme which accounts for pollution (dispersion) errors. Numerical results are given for problems in two and three space dimensions, to second and fourth-order accuracy, and they show the potential of the approach to solve a wide range of large-scale problems.

Paper Structure

This paper contains 32 sections, 5 theorems, 62 equations, 25 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Governing equations
  4. The WaveHoltz fixed-point iteration
  5. Solving the wave equation in complex geometry using overset grids
  6. Discretizing PDEs on overset grid
  7. Discretizing the wave equation
  8. Convergence of the WaveHoltz fixed-point iteration
  9. WaveHoltz convergence: continuous in space and time
  10. Fully discrete convergence analysis
  11. Explicit time-stepping
  12. Implicit time-stepping
  13. WaveHoltz iteration: acceleration and performance
  14. Deflation: accelerating WaveHoltz by removing some slowly converging eigenmodes
  15. Implicit time-stepping with a very large time-step
  16. ...and 17 more sections

Key Result

Theorem 1

Assume $\lambda_m\ne\omega$ are the eigenvalues of the problem in eq:eigBVP so that $\vert\beta(\lambda_m)\vert<1$ for all $\lambda_m$. The WaveHoltz fixed-point iteration has asymptotic convergence rate $\mu$ given by

Figures (25)

  • Figure 1: Gaussian source amongst multiple bodes. Left: overset grid (coarse version) consisting of a background blue grid and body fitted grids around each object. Middle: computed WaveHoltz solution using implicit time-stepping with $10$ time-steps per period and deflation. Right: WaveHoltz convergence history for the fixed-point iteration and GMRES accelerated iteration (further details are provided in subsequent sections).
  • Figure 2: Top: a three-dimensional overlapping grid for a quarter-cylinder in a box. Bottom left and right: component grids for the cylindrical and box grids in the unit cube parameter space. Interpolation points at the grid overlap are marked and color-coded for each component grid.
  • Figure 3: WaveHoltz filter function $\beta$ for ${N_p}=1$, ${N_p}=2$, and ${N_p}=3$ periods per time-interval.
  • Figure 4: Left: discrete filter function $\beta_d$ and continuous filter $\beta$ for $N_t=5$. Right: the adjusted $\lambda^i_{h,m}$ for implicit time-stepping versus $\lambda_{h,m}$ for varying number of time-steps $N_t$, $\Delta t=T/\Delta t$, for $\omega=1$.
  • Figure 5: Plots of $1-\beta(\lambda)$ for ${N_p}=1$, ${N_p}=2$, and ${N_p}=3$ periods per time-interval. The Krylov solvers operate on the matrix $A=I-S_h$ which has eigenvalues $1-\beta(\lambda_{h,m})$ of which representative values are shown with red x's.
  • ...and 20 more figures

Theorems & Definitions (5)

  • Theorem 1: WaveHoltz FPI Convergence Rate
  • Theorem 2: Fully Discrete Explicit WaveHoltz FPI Convergence
  • Theorem 3: Fully Discrete Implicit WaveHoltz FPI Convergence Rate
  • Theorem 4: WaveHoltz FPI convergence with deflation
  • Theorem 5