Advection Schemes

Used as a keyword argument advection_scheme to MOLFiniteDifference.

Upwind Scheme

UpwindScheme(approx_order = 1)

Changes the direction of the stencil based on the sign of the coefficient of the odd order derivative to be discretized. Scheme order can be increased by changing the approx_order keyword argument. For more information, see Wikipedia

WENO Scheme of Jiang and Shu

WENOScheme(epsilon = 1e-6)

A more stable scheme, 5th order accurate, which is a weighted sum of several different schemes, weighted based on the curvature of the solution at the point in question. More stable and tolerant of discontinuities, at the cost of solve complexity.

epsilonis a quantity used to prevent vanishing denominators in the scheme, defaults to 1e-6. Problems with a lower magnitude solution will benefit from a smaller value.

Problems which require this scheme may also benefit from a Strong-Stability-Preserving (SSP) solver.

Problems with first order derivatives which multiply one another will need to use this scheme over the upwind scheme.

Supports only first order derivatives, other odd order derivatives are unsupported with this scheme. At present does not support Nonuniform grids, though this is a planned feature.

Specified on pages 8-9 of this document