Adaptive time stepping for generalized convolution quadrature

Abstract

AbstractThrough the current work, we aim to improve the boundary element method (BEM) in acoustics. The proposed adaptivity is based on a time‐domain boundary element formulation. In this setting, an elegant method for solving the acoustic problem is through the transformation of the underlying partial differential equation on to the boundary of the domain as a space–time boundary integral equation. However, accurate modeling of temporal behavior is crucial, requiring careful consideration of the time step size to ensure numerical stability and capture the dynamics of the system accurately. The boundary integral equation in the time domain has a convolution in time and the generalized convolution quadrature method (gCQ) is introduced, providing a framework for numerically evaluating the convolution integral within the boundary integral equation. The gCQ method under consideration uses a higher order time stepping method from the Runge–Kutta family. The gCQ method allows for adaptive time stepping, enabling the refinement of the time step size to focus computational resources where they are most needed. The current work shows that an approach similar to that of the ordinary differential equations can be included in the gCQ to introduce an adaptive control of the time step size in the 3D boundary element formulation. Numerical experiments are conducted to validate the effectiveness of the proposed approach. The results demonstrate improved solution resolution, particularly in capturing steep changes and localized variations through the adaptive time stepping scheme. The proposed method shows promising performance, dynamically adjusting the time step size based on the evolving solution, thereby enabling efficient and accurate computations in BEM simulations of radiation phenomena.