Stability of Friedrichs's scheme in the maximum norm for hyperbolic systems in one space dimension

摘要

We are concerned with Friedrichs's scheme for an initial value problem ut(t, x) = A(t, x)ux(t, x), u(0, x) = u0(x), where u0(x) belongs to L∞, not to L2. We show that Friedrichs's scheme is stable in the maximum norm ·L∞, provided that the system is regularly hyperbolic and that the eigenvalues di(t, x) (i = 1,2,..., N) of the N XN matrix A(t, x) satisfy the conditions 1±λdi(t, x)⩾0 (i = 1,2,..., N), where λ is a mesh ratio.