<p>Given an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varvec{A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">A</mi> </mrow> </math></EquationSource> </InlineEquation>-stable rational approximation to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varvec{e}^{\varvec{z}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow> <mi mathvariant="bold-italic">z</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> of order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varvec{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">p</mi> </mrow> </math></EquationSource> </InlineEquation>, numerical procedures are suggested to time integrate abstract, well-posed IBVPs, with time-dependent source term <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varvec{f}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">f</mi> </mrow> </math></EquationSource> </InlineEquation> and boundary value <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varvec{g}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">g</mi> </mrow> </math></EquationSource> </InlineEquation>. These procedures exhibit the optimal order <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varvec{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">p</mi> </mrow> </math></EquationSource> </InlineEquation> and can be implemented by using just one single evaluation of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\varvec{f}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">f</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\varvec{g}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">g</mi> </mrow> </math></EquationSource> </InlineEquation> per step, i.e., no evaluations of the derivatives of data are needed, and are of practical use at least for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\varvec{p \le 6}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">p</mi> <mo mathvariant="bold">≤</mo> <mn mathvariant="bold">6</mn> </mrow> </math></EquationSource> </InlineEquation>. The full discretization is also studied and the theoretical results are corroborated by numerical experiments.</p>

Rational methods for abstract linear initial boundary value problems without order reduction

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2026年6月26日

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Carlos Arranz-Simón,
Begoña Cano,
César Palencia

摘要

Given an $\varvec{A}$ $A$ -stable rational approximation to $\varvec{e}^{\varvec{z}}$ $e^{z}$ of order $\varvec{p}$ $p$ , numerical procedures are suggested to time integrate abstract, well-posed IBVPs, with time-dependent source term $\varvec{f}$ $f$ and boundary value $\varvec{g}$ $g$ . These procedures exhibit the optimal order $\varvec{p}$ $p$ and can be implemented by using just one single evaluation of $\varvec{f}$ $f$ and $\varvec{g}$ $g$ per step, i.e., no evaluations of the derivatives of data are needed, and are of practical use at least for $\varvec{p \le 6}$ $p \leq 6$ . The full discretization is also studied and the theoretical results are corroborated by numerical experiments.