On some difference analogues of PDEs with a delay
No difference scheme for equation (2) seems to be definitely convergent
In our experiments the schemes of type A and B and mixed A-B behave less stable than C or D, especially close to the boundary errors for A and B are relatively big
The schemes C and D are comparable and the error expressions are reasonable
Our scheme of type D for equation (2) is easily adaptable to a parallel computing process. It is enough to divide the set E = [0, T] × [−b, b] into rectangles E = [0, T] × [b j , b j ,b j+1 ] (j = 0,..., m − 1) where −b = b0 < ... < b m = b, and then we can minimize functional F j defined on E like F on E. Using many processors one can achieve better results for meshes with smaller steps h t and h x . The exchange of data between neighbouring regions is minimal.
A proper division of the set E and the question of well-posedness of initialor initial-boundary-value problem for equation (3) is still open and demands some more extensive investigations with the use of multi-processor mashines.
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