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• W2044283205 abstract "It was generally expected that monotone schemes are oscillation-free for hyperbolic conservation laws. However, recently local oscillations were observed and usually understood to be caused by relative phase errors. In order to further explain this, we first investigate the discretization of initial data that trigger the chequerboard mode, the highest frequency mode. Then we proceed to use the discrete Fourier analysis and the modified equation analysis to distinguish the dissipative and dispersive effects of numerical schemes for low frequency and high frequency modes, respectively. It is shown that the relative phase error is of order <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper O left-parenthesis 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathcal O}(1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the high frequency modes <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u Subscript j Superscript n Baseline equals lamda Subscript k Superscript n Baseline e Superscript i xi j> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>u</mml:mi> <mml:mi>j</mml:mi> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>k</mml:mi> <mml:mi>n</mml:mi> </mml:msubsup> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>u_j^n=lambda ^n_k e^{ixi j}</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=xi almost-equals pi> <mml:semantics> <mml:mrow> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mo>≈<!-- ≈ --></mml:mo> <mml:mi>π<!-- π --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>xi approx pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, but of order <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper O left-parenthesis xi squared right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathcal O}(xi ^2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for low frequency modes (<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi almost-equals 0> <mml:semantics> <mml:mrow> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mo>≈<!-- ≈ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>xi approx 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). In order to avoid numerical oscillations, the relative phase errors should be offset by numerical dissipation of at least the same order. Numerical damping, i.e. the zero order term in the corresponding modified equation, is important to dissipate the oscillations caused by the relative phase errors of high frequency modes. This is in contrast to the role of numerical viscosity, the second order term, which is the lowest order term usually present to suppress the relative phase errors of low frequency modes." @default.
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• W2044283205 date "2009-01-28" @default.
• W2044283205 modified "2023-10-16" @default.
• W2044283205 title "Local oscillations in finite difference solutions of hyperbolic conservation laws" @default.
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• W2044283205 doi "https://doi.org/10.1090/s0025-5718-09-02219-4" @default.
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