Stability of difference approximations of dissipative type for mixed initial-boundary value problems. I

Abstract:

H. O. Kreiss, [2], has recently extended the stability theory of difference approximations to include the hyperbolic system \[ u_t = A{u_x},0 \leqq x,t\] , with $A$ a diagonal matrix. Appropriate boundary and initial conditions are given. The amplification matrix $\hat Q(\xi )$ need not be diagonal. However, he required that $|\hat Q(\xi )| \leqq 1$. We use certain results in matrix theory and Wiener-Hopf factorization to replace this restrictive assumption by certain reasonable assumptions on accuracy of $\hat Q(\xi )$ and smoothness of an associated positive-definite symmetric matrix. This technique will be important in half-space problems in many space variables since for such problems the amplification matrix will certainly not be diagonal.