Numerical Simulation of Kuramoto–Sivashinsky Equation Using Finite Line Method

The Finite Line Method (FLM) is a strong-form numerical method that incorporates all advantages of the Finite Difference Method, including its simple formulation, ease of understanding, and straightforward implementation. Additionally, FLM is easy to form high-order schemes in a unified way and exhibits the strong geometric adaptability characteristic of Meshless Methods. By utilizing a recursive principle, FLM can easily construct the higher-order spatial partial derivatives. The Kuramoto–Sivashinsky (KS) equation is a fourth-order nonlinear equation that describes spatiotemporal chaos in dissipative systems. This equation combines fourth-order dispersion, second-order diffusion, and nonlinear advection terms, making it challenging in its numerical modeling and simulation. In this paper, FLM is extended to solve the KS equation, and its performance is systematically assessed through multiple KS-type test examples, with accuracy and stability validated against analytical solutions, demonstrating its capability for handling high-order nonlinear dynamics and providing a reliable and efficient approach for complex partial differential equations.