SLERP-TVDRK (STVDRK) Methods for Ordinary Differential Equations on Spheres

TL;DR

This paper introduces SLERP-TVDRK methods, a new class of numerical integrators for differential equations on spheres that improve accuracy and simplicity by leveraging the sphere's exponential map and spherical linear interpolation.

Contribution

The paper proposes a novel class of integrators, STVDRK, that use the sphere's exponential map and SLERP, eliminating projections and enhancing accuracy over traditional methods.

Findings

Improved accuracy over typical projective RK methods.

Elimination of projection steps simplifies implementation.

Successful application to eikonal equation and p-harmonic flows.

Abstract

We mimic the conventional explicit Total Variation Diminishing Runge-Kutta (TVDRK) schemes and propose a class of numerical integrators to solve differential equations on a unit sphere. Our approach utilizes the exponential map inherent to the sphere and employs spherical linear interpolation (SLERP). These modified schemes, named SLERP-TVDRK methods or STVDRK, offer improved accuracy compared to typical projective RK methods. Furthermore, they eliminate the need for any projection and provide a straightforward implementation. While we have successfully constructed STVDRK schemes only up to third-order accuracy, we explain the challenges in deriving STVDRK-r for r \ge 4. To showcase the effectiveness of our approach, we will demonstrate its application in solving the eikonal equation on the unit sphere and simulating p-harmonic flows using our proposed method.