SP Documentation

1.0

Introduction

This library is to solve partial differential equations on sphere. It extends SpherePack library and make it easy to use in C++. It also implements some of the common partial differential equations such as Laplace equation, Chafe-Infante equation, Barotropic vorticity equation and two-dimensional baroclinic atmosphere equations.

Usage examples

Laplace equation on the sphere
$\begin{eqnarray*} \Delta \psi &=& f(\varphi, \lambda) \\ \Delta \psi &=& \frac{1}{cos\varphi}\frac{\partial}{\partial\varphi}cos(\varphi)\frac{\partial}{\partial\varphi}\psi+ \frac{1}{cos^2\varphi}\frac{\partial^2}{\partial\lambda^2}\psi\\ \end{eqnarray*}$
Chafe-Infante equation on the sphere
$\begin{eqnarray*} \frac{du}{dt} &=& \mu \Delta u - \sigma u + f (u) \\ u(x,y,t)|_{t=0} &=& u_0 \\ \end{eqnarray*}$
The Barotropic vorticity equation on the sphere
$\begin{eqnarray*} \frac{\partial \Delta \psi}{\partial t} + J(\psi, \Delta \psi) + J(\psi, l + h) + \sigma \Delta \psi - \mu \Delta^2 \psi &=& f(\varphi, \lambda) \\ \psi|_{t=0}=\psi_0 \end{eqnarray*}$
The two-dimensional baroclinic atmosphere equations on the sphere
$\begin{eqnarray*} \frac{\partial \Delta u_1}{\partial t} + J(u_1, \Delta u_1 + l + h) + J(u_2, \Delta u_2) + \frac{\sigma}{2} \Delta (u_1 - u_2) - \mu \Delta^2 u_1 &=& f(\phi, \lambda)\\ \frac{\partial \Delta u_2}{\partial t} + J(u_1, \Delta u_2) + J(u_2, \Delta u_1 + l + h) + \frac{\sigma}{2} \Delta (u_1 + u_2) - \mu \Delta^2 u_2 &-&\\ -\alpha^2 (\frac{\partial u_2}{\partial t} + J(u_1, u_2) - \mu_1 \Delta u_2 + \sigma_1 u_2 + g(\phi, \lambda)) &=& 0,\\ u_1|_{t=0}&=&u_{10}\\ u_2|_{t=0}&=&u_{20}\\ \end{eqnarray*}$