Stability Theory of Difference Approximations for Multidimensional Initial-Boundary Value Problems

A stability theory is developed for dissipative difference approximations to multidimensional initial-boundary value problems. The original differential problem should be strictly hyperbolic and the difference problem consistent with the differential one. An algebra of pseudo-difference operators is built and later used to prove the stability of the difference approximation with variable coefficients. In addition, stability of the Cauchy problem for weakly dissipative difference approximations with variable coefficients is proved. 0. Introduction. In the last two decades the mixed initial-boundary value problems for hyperbolic systems of partial differential equations were extensively and thoroughly studied. We should especially mention the classical work of Kreiss in [7] where he proved the basic a priori estimate for the strictly hyperbolic systems in the case of zero initial data, by constructing a pseudodifferential symmetrizer in the plane tangent to the boundary. This result was further generalized by Agranovich in [1] for a wider class of hyperbolic systems and extended by Rauch in [16] and Sarason in [17] to the case of nonzero initial data. Later Majda and Osher in [11] considered also characteristic boundaries. On the other hand an intelligent numerical solution of these problems requires a stability analysis of their difference approximation based on a suitable stability theory. Such a theory was, however, developed only in the one space dimensional case, first by Kreiss in [8] and Osher in [14] for some two-step dissipative schemes and nonzero initial conditions and then by Gustafsson, Kreiss and Sundström in [5] for more general dissipative as well as strictly nondissipative schemes with zero initial conditions. An attempt to generalize these results for multidimensional problems encounters two main difficulties. One is that the Fourier symbol of the difference operator, though consistent with the original differential equations, may approximate a very complicated hyperbolic-parabolic system when the dual variables of the symbol are not in a neighborhood of zero. Although such situations cannot be investigated in general, some difference schemes such as Lax-Friedrichs, modified Lax-Wendroff and Burstein schemes, for which the symbol is a polynomial of one matrix, could be analyzed completely. Analysis of that type is carried out in [13] for the Burstein scheme and characteristic boundary. The second difficulty occurs in the construction of the Kreiss symmetrizer for the difference problem when Received October 7, 1981. 1980 Mathematics Subject Classification. Primary 65N10, 65M10.