Type: Article
Publication Date: 1985-01-01
Citations: 21
DOI: https://doi.org/10.1090/s0025-5718-1985-0777269-7
New convenient stability criteria are provided in this paper for a large class of finite-difference approximations to initial-boundary value problems associated with the hyperbolic system <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold u Subscript t Baseline equals upper A bold u Subscript x Baseline plus upper B bold u plus bold f"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">u</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">u</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>x</mml:mi> </mml:msub> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>B</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">u</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">f</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {u}}_t} = A{{\mathbf {u}}_x} + B{\mathbf {u}} + {\mathbf {f}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the quarter plane <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x greater-than-or-slanted-equals 0"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>⩾<!-- ⩾ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">x \geqslant 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t greater-than-or-slanted-equals 0"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>⩾<!-- ⩾ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">t \geqslant 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Using the new criteria, stability is easily established for numerous combinations of well-known basic schemes and boundary conditions, thus generalizing many special cases studied in the recent literature.