Convergence theorem for difference approximations of hyperbolic quasilinear initial-boundary value problems

Abstract

Dissipative difference approximations to multi-dimensional hyperbolic quasi-linear initial-boundary value problems are considered. The difference approximation is assumed to be consistent with the differential problem and its linearization should be stable in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{l_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A formal asymptotic expansion to the difference solution is constructed. This expansion includes boundary and initial layers. It is proved that the expansion indeed approximates the difference solution to the required order. As a result, the difference solution converges to the differential one as the mesh size <italic>h</italic> tends to 0.