Computational PDEs: A Leap Forward

The Leapfrog Scheme for Hyperbolic PDEs

The leapfrog method, a significant numerical technique in computational partial differential equations (PDEs), has been widely used to solve hyperbolic PDEs. However, its stability and accuracy properties remain crucial to ensuring reliable simulations.

LTE Stability and Phase Error

The local truncation error (LTE) of the leapfrog scheme has a direct impact on its stability. For hyperbolic PDEs, the LTE exhibits a phase error, which can lead to numerical instability if not controlled. This phase error must be minimized for accurate solutions.

The Lax Theorem

The Lax theorem provides a fundamental criterion for the stability of the leapfrog scheme. It states that the scheme is stable if and only if a certain condition involving the Courant number and the wave propagation speed is satisfied. This condition ensures that the numerical solution remains bounded and does not amplify errors over time.

Understanding the stability and phase error of the leapfrog scheme is essential for successful numerical simulation of hyperbolic PDEs. By carefully considering these factors, researchers can develop robust and accurate algorithms that provide reliable scientific insights.