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• W2000995318 abstract "H. O. Kreiss, [2], has recently extended the stability theory of difference approximations to include the hyperbolic system <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u Subscript t Baseline equals upper A u Subscript x Baseline comma 0 less-than-over-equals x comma t> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>≦<!-- ≦ --></mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>u_t = A{u_x},0 leqq x,t</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> , with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a diagonal matrix. Appropriate boundary and initial conditions are given. The amplification matrix <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=ModifyingAbove upper Q With caret left-parenthesis xi right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo stretchy=false>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>hat Q(xi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> need not be diagonal. However, he required that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartAbsoluteValue ModifyingAbove upper Q With caret left-parenthesis xi right-parenthesis EndAbsoluteValue less-than-over-equals 1> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo stretchy=false>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mo>≦<!-- ≦ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>|hat Q(xi )| leqq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We use certain results in matrix theory and Wiener-Hopf factorization to replace this restrictive assumption by certain reasonable assumptions on accuracy of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=ModifyingAbove upper Q With caret left-parenthesis xi right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo stretchy=false>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>hat Q(xi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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." @default.
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• W2000995318 date "1969-01-01" @default.
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• W2000995318 title "Stability of difference approximations of dissipative type for mixed initial-boundary value problems. I" @default.
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• W2000995318 doi "https://doi.org/10.1090/s0025-5718-1969-0246530-8" @default.
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