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A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations.
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
For arbitrary stencil points and any derivative of order < up to one less than the number of stencil points, the finite difference coefficients can be obtained by solving the linear equations [6] ( s 1 0 ⋯ s N 0 ⋮ ⋱ ⋮ s 1 N − 1 ⋯ s N N − 1 ) ( a 1 ⋮ a N ) = d !
Therefore, the true derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line: ′ = (+) (). Since immediately substituting 0 for h results in 0 0 {\displaystyle {\frac {0}{0}}} indeterminate form , calculating the derivative directly can be unintuitive.
An illustration of the five-point stencil in one and two dimensions (top, and bottom, respectively). In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors".
A finite difference (FD) model of a differential equation (DE) can be formed by simply replacing the derivatives with FD approximations. But this is a naive "translation." If we literally translate from English to Japanese by making a one-to-one correspondence between words, the original meaning is often lost.
Figure 1.Comparison of different schemes. In applied mathematics, the central differencing scheme is a finite difference method that optimizes the approximation for the differential operator in the central node of the considered patch and provides numerical solutions to differential equations. [1]
The finite difference coefficients for a given stencil are fixed by the choice of node points. The coefficients may be calculated by taking the derivative of the Lagrange polynomial interpolating between the node points, [3] by computing the Taylor expansion around each node point and solving a linear system, [4] or by enforcing that the stencil is exact for monomials up to the degree of the ...