Abstract
This chapter highlights the rate of convergence of difference schemes for hyperbolic equations. It presents a survey of results on the rate of convergence of finite difference schemes applied to initial-value problems for hyperbolic equations. It discusses the dependence of this rate of convergence upon the smoothness of the initial values. The chapter discusses the spaces of functions in which the results are expressed and also states some interpolation properties of these spaces. It presents a general convergence result that applies to the case when the hyperbolic system is correctly posed and the difference operator is stable in the Lp space under consideration. As these assumptions are in general satisfied only in L2, their applicability to other Lp spaces is limited to special cases. The chapter presents a maximum-norm convergence result based on the L2-theory and on an embedding lemma of the Sobolev type. The chapter also explores the scalar one-dimensional case.
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