2024/10/14 by Shingyu Leung, Leung, Shingyu, Wai Ming Chau +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2410.10420
openalex publication_date 2024/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ≥ 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.