Approximation of solutions to non-stationary Stokes system

TL;DR

This paper introduces a fast, high-order semi-analytic method for approximating solutions to the non-stationary Stokes system in three dimensions, utilizing harmonic potential and heat equation cubature formulas.

Contribution

It develops a novel high-order approximation technique combining semi-analytic cubature formulas for the harmonic potential and heat equations, with proven convergence orders up to 8.

Findings

Achieves high accuracy with convergence orders 2, 4, 6, and 8.

Provides fast semi-analytic cubature formulas for harmonic potentials.

Demonstrates effectiveness through numerical experiments.

Abstract

We propose a fast method for high order approximations of the solution of the Cauchy problem for the linear non-stationary Stokes system in $R^3$ in the unknown velocity $\bf u$ and kinematic pressure $P$. The density ${\bf f}({\bf x},t)$ and the divergence vector-free initial value ${\bf g }(\bf x)$ are smooth and rapidly decreasing as $|\bf x|$ tends to infinity. We construct the vector $\bf u$ in the form $\bf u=\bf u_1+\bf u_2$ where $\bf u_1$ solves a system of homogeneous heat equations and $\bf u_2$ solves a system of non-homogeneous heat equations with right-hand side ${\bf f}-\nabla P$. Moreover $P=-{\cal L}( \nabla\cdot \bf f)$ where $\cal L$ denotes the harmonic potential. Fast semi-analytic cubature formulas for computing the harmonic potential and the solution of the heat equation based on the approximation of the data by functions with analitically known potentials are considered. In addition, the gradient $\nabla P$ can be approximated by the gradient of the cubature of $P$, which is a semi-analytic formula too. We derive fast and accurate high order formulas for the approximation of $\bf u_1,\bf u_2$, $P$ and $\nabla P$. The accuracy of the method and the convergence order $2,4,6$ and $8$ are confirmed by numerical experiments.