Numerical Analysis

An implicit scheme, invented b John Crank (1916-) and Phyllis Nicolson (1 917-1968), is based on numerical approximations for solutions at the point that lies between the rows in the grid. Specifically, the approximation used for is obtained from the central-difference formula, .

As examples of elliptic partial differential equations , we consider the Laplace equation , Poisson equation , and Helmholtz equation . Recall that the Laplacian of the function u(x,y) is . With this notation, we can write the Laplace, Poisson, and Helmholtz equations in the following forms: It is often the case that the boundary values for the function u(x,y) are known at all points on the sides of a rectangular region R in the plane. In this case, each of these equations can be solved by the numerical technique known as the finite-difference method.