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

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.

Figures (10)

• Figure 1: (Section \ref{['SubSec:Stability']} Stability Condition Due to SLERP) We assume the motion is defined on the great circle of the sphere with the initial location on the south pole (i.e., the orange dot). (a) If the initial location travels more than $h\| f(\mathbf{p},t) \|>\pi$, the geodesic to the final location (the red dot) will be given by the segment indicated by the blue curve instead of the segment of travel. (b) For STVDRK2, we assume $\mathbf{p}^n$ (the orange dot) travels to $\mathbf{q}_1$ (the dark orange dot) and then $\mathbf{q}_2$ (the red dot). The midpoint along the geodesic from $\mathbf{p}^n$ to $\mathbf{q}_2$ is given by the blue dot instead of the expected location indicated by the green square.
• Figure 2: (Section \ref{['SubSec:ExConvergence']}) (a) The $E_2$ errors in the solution obtained by SFE demonstrate first-order accuracy, while those obtained by TVDRK2, PTVDRK2, PTVDRK2', PTVDRK3', and our proposed STVDRK2 demonstrate second-order accuracy. (b) The $E_2$ errors in the solutions obtained by TVDRK3, PTVDRK3, and our proposed STVDRK3 exhibit third-order convergence. (c) The $E_2$ errors in the solutions obtained by STVDRK4, SSSPRK(5,4), and SSSPRK(10,4) when the convex combination is done using progressive SLERP. We compare these errors with the one computed using STVDRK3. All schemes exhibit only third-order convergence. (d) The $E_2$ errors in the solutions obtained by STVDRK4, SSSPRK(5,4) and SSSPRK(10,4) when the convex combination is done using the Fréchet mean. All schemes exhibit only second-order convergence. (e) The $E_2$ errors in the solutions obtained by Radau-IIA(3), DAE3 and DASSL methods. We have also shown the error in the STVDRK3 solutions as a reference. (f) The $E_2$ errors in the solutions obtained by high-order methods DOP54 and DAE5. We have also shown the error in the STVDRK3 solutions as a reference.
• Figure 3: (Section \ref{['SubSec:ExConvergence']}) (a) $E_{\text{norm}}$ errors in the solutions obtained by some explicit RK methods including TVDRK2, TVDRK3, RK3 and RK4. They demonstrate third or fourth-order convergence. (b) $E_{\text{norm}}$ errors in the solutions obtained by some popular methods including Radau-IIA(3), DOP54 and DAE methods. They demonstrate third or fifth-order convergence.
• Figure 4: (Section \ref{['Ex:Stability']}) The numerical solutions obtained using (a) SFE, (b) STVDRK2, and (c) STVDRK3 are computed with time steps that both (i) satisfy and (ii) violate the A-stability condition. For the SFE and STVDRK2 methods, we adopt step sizes of 1.99 and 2.01. In the case of STVDRK3, we choose step sizes of 2.51 and 2.52.
• Figure 5: (Section \ref{['SubSec:SurfaceEikonal']} with $v(\mathbf{x})=1$) $L_2$-errors in the solutions at $t=\pi/2$ obtained by various numerical integrators.

Theorems & Definitions (3)

• Proposition 1
• Theorem 1
• proof