The Scope of the Energy Method

where Uj and Vj denote the values of the approximate solution at position jA and at times t and / + A respectively. We have chosen here for simplicity the same value A for both time and space increments. The coefficients Cj depend on the index k ; we assume that this dependence is Lipschitz continuous, i.e., that c(k + l)— c(k) = 0(A). Since the differential equation (1) is homogeneous, u = const, is a solution. We assume, as part of the consistency of (1) and (2), that u — const, also is a solution of (2), i.e., that