PDESolver Introduction

PDESolver Documentation

Welcome to the PDESolver documentation. These documents provide an overview of PDESolver, a Julia based solver for partial differential equations. This page will describe the form of the equations and the Summation-By-Parts operators used to discretize them. The Table of Contents links to the components of PDESolver that calculate each term.

Form of the Equation

PDESolver discretizes equation in the form:

\[\frac{\partial q}{\partial t} = \mathcal{R}(q, t)\]

where $q$ is the variable being solved for, $\mathcal{R}(u, t)$ is called the residual and $t$ is time. The residual contains all the spatial derivatives. For example, the residual for the 1D advection equation is $ a\frac{\partial q}{\partial x}$, where $a$ is the advection velocity. The code is capable of solving both unsteady problems, in the form shown above, and steady problem of the form

\[\mathcal{R}(q) = 0.\]

The code is structured such that the physics modules are responsible for evaluating the residual and the Nonlinear Solvers are responsible for the time term in the unsteady case, or solving the nonlinear rootfinding problem for steady cases.

Summation-by-Parts operators

Summation-by-Parts (SBP) operators are used to discretize the spatial derivatives in the residual. In particular, we use the multi-dimensional Summation-by-Parts operators first defined in 

  Hicken, J.E., Del Rey Fernandez, D.C, and Zingg, D.W., "Multi-dimensional 
  Summation-by-Parts Operators: General Theory and Application to 
  Simplex Elements", SIAM Journal on Scientific Computing, Vol, 38, No. 4, 2016,
   pp. A1935-A1958

See the introductory PDF (to be posted shortly) for an introduction to the operators.

Table of Contents