High-order implicit Runge-Kutta Fourier pseudospectral methods for wave equations

The dispersion error, also known as the pollution effect, is one of the main difficulties in numerical solutions to the wave propagation problem at high wavenumbers. The pollution effect, especially in mesh-based methods, can potentially be controlled by using either finer meshes or higher-order discretizations. Using finer meshes often leads to large systems that are computationally expensive to solve, especially for medium to high wavenumbers. Therefore, higher-order approximations are preferred to achieve good accuracy with manageable complexity. In this work, we will present high-order methods with implicit Runge-Kutta time integration and Fourier pseudospectral spatial approximations for the wave equation in a domain of interest surrounded by a sponge layer. At each time step, applying an appropriate A-stable implicit Runge-Kutta time-stepping method results in a modified Helmholtz equation that needs to be solved, for which an efficient iterative functional evaluation method with Fourier pseudospectral approximations will be proposed. The functional evaluation method transforms the equation into a functional iteration problem associated with an exponential operator that can be solved iteratively with guaranteed efficient convergence, where the exponential operator is evaluated by high-order operator splitting techniques and Fourier pseudospectral approximations. Numerical experiments are performed to demonstrate the effectiveness of the proposed method.