Accuracy Barriers of Two Time Level Difference Schemes for Hyperbolic Equations

Abstract

For two time level schemes for $u_1 = cu_x $ we give for each noninteger Courant number $\mu $ a bound for the order of a stable scheme. If this bound is attained we give a lower bound for the error constant and give the scheme which attains these bounds. All these barriers to the accuracy of stable schemes are expressed in a physical fashion in terms of the number of upwind and downwind points. This notation reveals a certain symmetry with respect to a characteristic. The proofs are carried out by comparing a general scheme to the “optimal” one using order stars.