A Semi-Lagrangian Spherical Essentially Non-Oscillatory (SENO) Scheme for Advection Equations of S2-valued Functions

TL;DR

This paper introduces a semi-Lagrangian SENO scheme for accurately solving advection equations of spherical-valued functions, effectively handling discontinuities and reducing oscillations.

Contribution

The paper extends semi-Lagrangian methods with SENO interpolation to efficiently solve advection equations for $ ext{S}^2$-valued functions, addressing discontinuities.

Findings

The scheme accurately models the evolution of $ ext{S}^2$-valued functions.

SENO interpolation reduces spurious oscillations in high-order reconstructions.

Numerical examples demonstrate the method's effectiveness and accuracy.

Abstract

We develop a numerical scheme for solving the advection equation of $\mathbb{S}^2$-valued functions of real variables, which models the time-evolution of a $\mathbb{S}^2$-valued mapping on the real line by a known velocity field. The idea is to extend the semi-Lagrangian method for the linear scalar advection equation. We first construct the backward flow map between two adjacent time levels and then interpolate the discrete ordered data of $\mathbb{S}^2$. To handle $\mathbb{S}^2$-functions which have kinks or sharp discontinuity in their components, we incorporate the \textit{Spherical Essentially Non-Oscillatory} (SENO) interpolation method, which effectively reduces the spurious oscillations in high-order reconstructions. We will show multiple examples to demonstrate the accuracy and effectiveness of the proposed algorithm for the partial differential equation of $\mathbb{S}^2$-functions.