# Spatial Derivatives

Default 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:

First order derivatives:
dx3c(U): Three point centered
dx5c(U): Five point centered (also the default dx(U))
dx7c(U): Seven point centered
dx9c(U): Nine point centered
dx11c(U): Eleven point centered

U is the variable or expression for which a spatial derivative is desired

First order derivatives - upwind:
dxu2(U,v): Two point upwind
dxu3(U,v): Three point upwind
dxu5(U,v): Five point upwind
dxu4b(U,v): Four point biased upwind
dxu5b(U,v): Five point biased upwind
dxu25(U,v): Five point biased upwind (smooth ends) (also the default dxu(U,v)

U is the variable or expression for which a spatial derivative is desired
v is the constant or f(t) whose sign determines the upwind direction

Second order derivatives:
dxx3c(U,ab): Three point centered
dxx5c(U,ab): Five point centered (also the default dxx(U,ab)

U is the variable or expression for which a spatial derivative is desired
ab is a boundary type indicator. It could have the values DD, DN, NN, or ND (D = Dirichlet, N = Neumann)

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