Multigrid Methods for Solving Hyperbolic PDEs with Rough Coefficients

In this article, we present a new method for solving wave problems that combines multiple scales and high-order polynomials to achieve better accuracy and efficiency. The method is called the p-LOD-θ method and it’s based on the idea of using a hierarchy of grids with different resolutions to represent the solution of the wave equation.
The key insight of the method is that the wave equation can be written in a form that separates the fast and slow variables, allowing us to use high-order polynomials to represent the slow variables while keeping the fast variables simple. This leads to a significant reduction in computational cost without sacrificing accuracy.
We demonstrate the effectiveness of the p-LOD-θ method through several examples, including the simulation of wave propagation in a rectangular domain and the solution of the wave equation with rough coefficients. The results show that our method outperforms traditional methods in terms of both accuracy and efficiency, making it a valuable tool for solving wave problems in various fields.
By using a hierarchy of grids with different resolutions, we can represent the solution of the wave equation at multiple scales, allowing us to capture both the large-scale features of the wave field and the small-scale details. This makes our method particularly useful for problems where the wave field has multiple scales, such as in geophysics or fluid dynamics.
In addition, we show that our method is energy-conservative, which means that the total energy of the wave field is preserved over time, a crucial property for any numerical method.
Overall, the p-LOD-θ method provides a powerful tool for solving wave problems by combining the advantages of high-order polynomials and multiscale methods. Its ability to capture both large-scale and small-scale features of the wave field makes it particularly useful for problems with multiple scales, and its energy conservation properties ensure that the total energy of the wave field is preserved over time.