An account is given of the methods available to improve the accuracy of arithmetical solutions on a square mesh for two-dimensional partial differential equations. ‘Squaring’ (1) and ‘Relaxation’ (5) depend for their accuracy on the fineness of the mesh on which the finite difference equations replace the differential equations. The same accuracy can be obtained on a coarse mesh, with less labour, by the use of more accurate difference formulae, rather than by refining the mesh. There are two ways of doing this: (a) by using a higher order difference equation as a closer approximation to the differential equation, or (b) by calculating a correction term from the higher differences to apply to the results obtained by using the first approximation. A new method of calculating the correction term is developed.
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L. C. Woods (1950) studied this question.