Schemes for the heat conduction equation with several spatial variables

3 SCHEMES FOR THE HEAT CONDUCTION EQUATION WITH SEVERAL SPATIAL VARIABLES 

The explicit difference scheme. The schemes considered in Section 1 may be generalized to the case of the heat conduction equation with several spatial variables. 

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The starting point in the further development of the difference scheme is the approximation of the elliptic operator Aw. We learn from Section 1 of Chapter 4 that at all of the inner nodes Aw Aw for x G 

By inserting in (1) the difference operator A in place of the Laplace operator we are led to the system of differential-difference equations 

For the purposes of the present section, let us introduce the grid uih = xt G (5 in G and denote by -yh the set of all nodes of uih belonging to T and by wh the set of all inner nodes xi G, so that wh = wh + jh-The starting point in the further development of the difference scheme is the approximation of the elliptic operator Ati. We learn from Section 1 of Chapter 4 that at all of the inner nodes An for x G LOh. 

Common practice involves the decomposition for the solution of the problem concerned as a sum y = y + y, where y is a solution to the homogeneous equation 

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Schemes for the heat conduction equation with several spatial variables... 

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