This page describes the implementation of dissipation-based continuation (DBC) in OPTIMA2D.

DBC Modifications To the PROBE Flow Solver

We wish to solve the non-linear problem

$R ( \widehat{Q}) = 0$

To find a solution Q^ that satisfies the above equation, Newton's method is used:

$\mathcal{A}^{(n)}\Delta \widehat{Q}^{(n)} = -R^{(n)}$

where

$\mathcal {A}^{(n)} = \left( \frac{\partial{R}}{\partial{\widehat{Q}}} \right)^{(n)}$

The Newton update is then calculated as

$\widehat{Q}^{(n+1)} = \widehat{Q}^{(n)} + \Delta \widehat{Q}^{(n)}$

Right Hand Side Modifications:

The right hand side (RHS) of the linear system arising at each Newton-Krylov iteration is -R, where R is the residual of the discretized governing equations. For example, the residual of the discretized Euler equations is given by

$R ( \widehat{Q}) = \delta_{\xi} \widehat{E} + \delta_{\eta} \widehat{F} - \nabla_{\xi}D_{\xi} - \nabla_{\eta}D_{\eta}$

Let H be the modified residual given by

$H ( \widehat{Q}) = R ( \widehat{Q}) + \lambda \hspace{0.03in} D ( \widehat{Q})$

The following subroutines are affected:

BCRHS
VBCRHS
RHSX
RHSY
COEF24X
COEF24Y
FILTERX
FILTERY
PROBE

Left Hand Side Modifications:

Input Parameters for Controlling DBC Performance:

DISSCON
LAMDISSMAX
DCTOL
LAMKILL