Generally Unconditionally Stable Difference Operators

Abstract

u(x, 0) = uo(x), where P(D) is a partial differential operator of order p with respect to x with constant matrix coefficients. We shall study consistent difference operators of the form (2) v(x, t + k) = Eh,kv(x, t) = f(kP(ah))v(x, t), where f(z) is a rational function, h and k are the mesh-widths in x and t, respectively, and where ahV = (ah,lV, , ah,V) with NOhzv(x) = (ih)<'Eljjj, d(j)v(x + jhel) is consistent with Dv = (i7'av/0x1,, i-1a/v9xd) and the ah,N, I = 1, ***, d, have the same real-valued symbol d(rq) = i-'Ej d(j exp (iji). We say that such an operator is generally unconditionally stable if it is unconditionally stable for all correctly posed initial-value problems. Depending on whether the correctness of (1) is interpreted in Petrowsky's sense or in the sense of Lax and Richtmyer, the stability is understood to be in the sense of Forsythe and Wasow or Lax and Richtmyer, respectively. It is proved that in both cases the necessary and sufficient condition for general unconditional stability is