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• W2004759358 abstract "The stiffness in some systems of nonlinear differential equations is shown to be characterized by single stiff equations of the form <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=y prime equals g prime left-parenthesis x right-parenthesis plus lamda left-brace y minus g left-parenthesis x right-parenthesis right-brace period> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>g</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>y</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>y’ = g’(x) + lambda {y - g(x)}.</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> The stability and accuracy of numerical approximations to the solution <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=y equals g left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>y = g(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, obtained using implicit one-step integration methods, are studied. An <italic>S</italic>-stability property is introduced for this problem, generalizing the concept of <italic>A</italic>-stability. A set of stiffly accurate one-step methods is identified and the concept of stiff order is defined in the limit <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R e left-parenthesis negative lamda right-parenthesis right-arrow normal infinity> <mml:semantics> <mml:mrow> <mml:mi>Re</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {Re}(-lambda ) to infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. These additional properties are enumerated for several classes of <italic>A</italic>-stable one-step methods, and are used to predict the behaviour of numerical solutions to stiff nonlinear initial-value problems obtained using such methods. A family of methods based on a compromise between accuracy and stability considerations is recommended for use on practical problems." @default.
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• W2004759358 date "1974-01-01" @default.
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• W2004759358 title "On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations" @default.
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• W2004759358 doi "https://doi.org/10.1090/s0025-5718-1974-0331793-2" @default.
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