Lyapunov-Sylvester operators for Kuramoto-Sivashinsky Equation
This provides a new numerical approach for a specific nonlinear PDE, but the impact is incremental as it applies existing operator techniques to a known problem.
The authors develop a numerical method using generalized Lyapunov-Sylvester operators to solve the two-dimensional Kuramoto-Sivashinsky equation, proving stability and convergence. Numerical tests validate the theoretical results.
A numerical method is developed leading to algebraic systems based on generalized Lyapunov-Sylvester operators to approximate the solution of two-dimensional Kuramoto-Sivashinsky equation. It consists of an order reduction method and a finite difference discretization which is proved to be uniquely solvable, stable and convergent by using Lyapunov criterion and manipulating generalized Lyapunov-Sylvester operators. Some numerical implementations are provided at the end to validate the theoretical results.