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<p>In this paper, we developed novel fourth-order Runge-Kutta type exponential time differencing (ETD) <i>A</i>-stable and <i>L</i>-stable methods for space-fractional nonlinear reaction-diffusion equations with initial non-smooth or smooth data. Based on compact finite differences, a fourth-order technique is used for spatial discretization, while ETD is employed to discretize the time. Our novel numerical schemes have the benefit of explicitly handling the nonlinear term. The well-known issue of numerical instability related to computing the matrix exponential is addressed using the real single-pole rational approximation, namely the restricted Padé approximation approach. The corresponding ETD-<i>A</i>-stable and <i>L</i>-stable methods are obtained. Convergence, error estimates, and stability analysis of the suggested approaches are studied theoretically. Under a global Lipschitz continuity assumption, the unconditional <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> numerical stability is established. Moreover, the convergence order of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathcal {O}}(k^4)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mn>4</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for the derived methods is also studied in the norm <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>. Numerical experiments demonstrate the advantages of the methods in computational accuracy, efficiency, and reliability.</p>

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High-Order Time-Stepping Methods for Two-Dimensional Space-Fractional Reaction-Diffusion Models

  • 期刊论文

  • 发表日期：

    2025年12月9日

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  • Shahzad Sarwar

摘要

In this paper, we developed novel fourth-order Runge-Kutta type exponential time differencing (ETD) A-stable and L-stable methods for space-fractional nonlinear reaction-diffusion equations with initial non-smooth or smooth data. Based on compact finite differences, a fourth-order technique is used for spatial discretization, while ETD is employed to discretize the time. Our novel numerical schemes have the benefit of explicitly handling the nonlinear term. The well-known issue of numerical instability related to computing the matrix exponential is addressed using the real single-pole rational approximation, namely the restricted Padé approximation approach. The corresponding ETD-A-stable and L-stable methods are obtained. Convergence, error estimates, and stability analysis of the suggested approaches are studied theoretically. Under a global Lipschitz continuity assumption, the unconditional \(L^2\) L 2 numerical stability is established. Moreover, the convergence order of \({\mathcal {O}}(k^4)\) O ( k 4 ) for the derived methods is also studied in the norm \(L^2\) L 2 . Numerical experiments demonstrate the advantages of the methods in computational accuracy, efficiency, and reliability.