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• W1967053584 abstract "Many second order accurate nonoscillatory schemes are based on the minmod limiter, e.g., the Nessyahu–Tadmor scheme. It is well known that the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Subscript p> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>L_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-error of monotone finite difference methods for the linear advection equation is of order <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1 slash 2> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>1/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for initial data in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W Superscript 1 Baseline left-parenthesis upper L Subscript p Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>W</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>W^1(L_p)</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=1 less-than-or-equal-to p less-than-or-equal-to normal infinity> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>1leq pleq infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For second or higher order nonoscillatory schemes very little is known because they are nonlinear even for the simple advection equation. In this paper, in the case of a linear advection equation with monotone initial data, it is shown that the order of the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L 2> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>L_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-error for a class of second order schemes based on the minmod limiter is of order at least <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=5 slash 8> <mml:semantics> <mml:mrow> <mml:mn>5</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>8</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>5/8</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in contrast to the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1 slash 2> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>1/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> order for any formally first order scheme." @default.
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• W1967053584 date "2006-05-23" @default.
• W1967053584 modified "2023-10-17" @default.
• W1967053584 title "Order of convergence of second order schemes based on the minmod limiter" @default.
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