Richard Courant 
On the Partial Difference Equations, of Mathematical Physics [PDF ebook] 

Whilst the greatest effort has been made to ensure the quality of this text, due to the historical nature of this content, in some rare cases there may be minor issues with legibility. Nyo-7689 of arbitrarily high order tends to the corresponding deriva tive. Nowhere do we assume the existence of the solution to the differential equation problem on the contrary, we obtain a simple existence proof by using the limiting processi. For the case of elliptic equations convergence is obtained independently of the choice of mesh, but we will find that for the case of the initial value problem for hyperbolic equations, convergence is obtained only if the ratio of the mesh widths in different directions satis fies certain inequalities which ih turn depend on the position of the characteristics relative to the mesh.