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Backward finite difference [ edit ] To get the coefficients of the backward approximations from those of the forward ones, give all odd derivatives listed in the table in the previous section the opposite sign, whereas for even derivatives the signs stay the same.
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The order of differencing can be reversed for the time step (i.e., forward/backward followed by backward/forward). For nonlinear equations, this procedure provides the best results. For linear equations, the MacCormack scheme is equivalent to the Lax–Wendroff method. [4]
In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for f ′(x + h / 2 ) and f ′(x − h / 2 ) and applying a central difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the second derivative of f:
Forward-Backward Euler method The result of applying both the Forward Euler method and the Forward-Backward Euler method for = and =. In order to apply the IMEX-scheme, consider a slightly different differential equation:
Finite-difference method can contain the standard explicit 2nd order scheme for heat equation, some analysis (consistency, stability, convergence), other schemes like implicit in time and Crank-Nicholson (up to here this is the content of Finite difference#Example: the heat equation), discussion on how to handle boundary conditions, the ...
The method is based on finite differences where the differentiation operators exhibit summation-by-parts properties. Typically, these operators consist of differentiation matrices with central difference stencils in the interior with carefully chosen one-sided boundary stencils designed to mimic integration-by-parts in the discrete setting.
In my view, and as explained after the cited text, given a certain mesh size , a central difference is not necessarily more accurate than a forward or backward difference. What seems true is that the rate of convergence to the actual derivative in the limit for small h {\displaystyle h} is faster.