On the Stability of Friedrichs' Scheme and the Modified Lax-Wendroff Scheme*

Necessary and sufficient stability criteria for Friedrichs' scheme and the modified Lax-Wendroff scheme with smooth coefficients are derived by means of Kreiss' Matrix Theorem and the first Stability Theorem of Lax and Nirenberg. In this note we derive necessary and sufficient stability criteria for Friedrichs' scheme and the modified Lax-Wendroff scheme [8] for the hyperbolic system n (1) ". = X) a,(*)«*, i-i of first-order differential equations with variable coefficient matrices. Friedrichs' scheme v(x, t + k) = Shv(x, t) is given by the difference operator Sh = ~ Z ITi + 771] + |x £ ai{x)\T, T¡1\ = C„ + XAk. with Tj representing the translation operator, by the amount h in the xt direction. The symbol of Sh is the matrix (2) s(x, O = ( £ cos {, )/ + iX £ fl/W sin £, a eg)/ + iXa(x, sin 0, where X = k/h, k being the time step and h the space mesh size, $ = {£„ • • • , £,}, sin ? = {sin |i, • • • , sin £„}, and / is the identity matrix. With the above notation, the modified Lax-Wendroff scheme [8] is given by v(x, t + k) = Lhv(x, i), where the operator Lh= I + \Ak[Ch + h\Ak] Received May 26, 1969, revised December 1, 1969. AMS 1969 subject classifications. Primary 6567; Secondary 3553, 3523.