Spatial DerivativesDefault approximations: dx (u) : First spatial (x) derivative of any expression u with x-dependence. By default, first derivatives are approximated by a five point centered approximation. dxu (u,c): First upwind spatial (x) derivative of any expression u with x-dependence. By default, first upwind derivatives are approximated by a five point biased upwind approximation (smooth ends). Upwind derivatives are recommended for strongly convective (hyperbolic) problems. c indicates the direction of upwind (a positive or negative value) dxx (u,ab): Second spatial (x) derivative of any expression with x-dependence. By default, second derivatives are approximated by five point centered approximations. ab denotes the type of lower and upper boundary condition. Possible values for a and b are D (for Dirichlet) and N (for Neumann.) Thus, ab could adopt the values of DD, DN, NN, or ND. Instead of the default approximations for spatial derivatives (dx, dxu, and
dxx), the following approximations may also be used: |
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