Stabilizing Explicit Schemes with Hyperviscosity and Filtering

This chapter presents a detailed and structured exploration of techniques used to enhance the stability of explicit finite-difference schemes, particularly in the presence of high-gradient flows or sharp solution features. It begins by examining how numerical instabilities arise from truncation errors and demonstrates how these errors disproportionately affect high-frequency components of the solution. To address this, the chapter introduces hyperviscosity introduced via a fourth-order artificial dissipation term into the diffusion equations. This technique selectively damps non-physical, high-wavenumber oscillations while preserving the integrity of large-scale dynamics. Through careful mathematical formulation and Fourier analysis, readers gain insight into how hyperviscosity modifies dispersion and dissipation behavior and how its parameters can be tuned for optimal stabilization in explicit schemes. In addition, the chapter explores Gaussian filtering as a complementary stabilization approach. Applied to the Laplacian term, this method smooths numerical noise while maintaining overall solution accuracy. Together, these tools provide a practical foundation for constructing robust and stable numerical solvers for diffusion-type problems and high-gradient flow simulations.