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Implicit Runge–Kutta and spectral Galerkin methods for Riesz space fractional/distributed-order diffusion equation

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Abstract

A numerical method with high accuracy both in time and in space is constructed for the Riesz space fractional diffusion equation, in which the temporal component is discretized by an s-stage implicit Runge–Kutta method and the spatial component is approximated by a spectral Galerkin method. For an algebraically stable Runge–Kutta method of order p\((p\ge s+1)\), the unconditional stability of the full discretization is proven and the convergence order of \(s+1\) in time is obtained. The optimal error estimate in space, with convergence order only depending on the regularity of initial value and f, is also derived. Meanwhile, this kind of method is applied to the Riesz space distributed-order diffusion equation, and similar stability and convergence results are obtained. Finally, numerical experiments are provided to illustrate the theoretical results.

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Abbreviations

\( {}_{-1}^{}D^{\mu }_{x}\) :

Left Riemann–Liouville fractional derivative operator with \(\mu >0\)

\((-{\varDelta })^{\frac{\alpha }{2}}\) :

Fractional Laplace operator with \(\alpha \in (0,1)\cup (1,2)\)

\(\frac{\partial ^{\alpha }}{\partial {\vert x\vert }^{\alpha }}\) :

Riesz fractional derivative operator with \(\alpha \in (0,2)\)

\({\hat{H}}\) :

Hilbert transform operator

\({\varLambda }\) :

\({\varLambda }=(-1,1)\)

\(\langle \cdot ,\cdot \rangle _{\omega ^{a,b},{\varOmega }}\) :

Inner product in \(L^{2}_{\omega ^{a,b}}({\varOmega })\)

\(\left\langle \cdot ,\cdot \right\rangle _{H^{k}({\varLambda })}\) :

Inner product in \(H^{k}({\varLambda })\) with \(k\in {\mathbb {N}}\)

\(\left\langle \cdot ,\cdot \right\rangle _{L^{2}({\varLambda })}\) :

Inner product in \(L^{2}({\varLambda })\)

\({\varOmega }\) :

\({\varOmega }=(\lambda _{1},\lambda _{2})\)

\(\omega ^{a,b}(x)\) :

\(\omega ^{a,b}(x) = (\frac{2}{\lambda _{2} - \lambda _{1}})^{a+b}(\lambda _{2} - x)^{a}(x - \lambda _{1})^{b}\) is the shifted Jacobi weight function with \(a, b>-1\)

\(\Vert \cdot \Vert _{\omega ^{a,b},{\varOmega }}\) :

Norm in \(L^{2}_{\omega ^{a,b}}({\varOmega })\)

\(\Vert \cdot \Vert _{H^{\mu }_{0}({\varLambda })}\) :

Norm in \(H^{\mu }_{0}({\varLambda })\) with \(\mu >0\)

\(\vert \cdot \vert _{H^{\mu }_{0}({\varLambda })}\) :

Semi-norm in \(H^{\mu }_{0}({\varLambda })\) with \(\mu >0\)

\(\Vert \cdot \Vert _{H^{k}({\varLambda })}\) :

Norm in \(H^{k}({\varLambda })\) with \(k\in {\mathbb {N}}\)

\(\Vert \cdot \Vert _{L^{2}({\varLambda })}\) :

Norm in \(L^{2}({\varLambda })\)

\(C^{\infty }({\varOmega })\) :

Space of infinitely differentiable functions on \({\varOmega }\)

\(H^{\mu }_{0}({\varLambda })\) :

Fractional derivative space on \({\varLambda }\) with \(\mu >0\)

\(H^{k}({\varLambda })\) :

Sobolev space on \({\varLambda }\) with \(k\in {\mathbb {N}}\)

\(H^{k}_{\omega ^{a,b},*}({\varOmega })\) :

\(H^{k}_{\omega ^{a,b},*}({\varOmega }) := \left\{ \eta (x) \in H^{1}{({\varOmega })} {\Big \vert } \frac{{\mathrm {d}}^{j}}{{\mathrm {d}} x^{j}}\eta (x) \in L^{2}_{\omega ^{a+j-1,b+j-1}}({\varOmega }),1 \le j \le k \right\} \) with \(k\in {\mathbb {N}}\)

\(H^{k}_{\omega ^{a,b}}({\varOmega })\) :

\(H^{k}_{\omega ^{a,b}}({\varOmega }):=\left\{ \eta (x){\Big \vert } \frac{{\mathrm {d}}^{j}}{{\mathrm {d}} x^{j}}\eta (x)\in L^{2}_{\omega ^{a+j,b+j}}({\varLambda }),~0\le j\le k \right\} \) with \(k\in {\mathbb {N}}\)

\(L^{2}({\varLambda })\) :

\(L^{2}\) space on \({\varLambda }\)

\(L^{2}_{\omega ^{a,b}}({\varOmega })\) :

\(L^{2}_{\omega ^{a,b}}({\varOmega }):=\{\eta (x)~\vert ~\eta (x)\) is measurable and \(\Vert \eta (x)\Vert _{\omega ^{a,b},{\varOmega }}<\infty \}\)

\(P_{N}({\varLambda })\) :

Space of polynomials defined on \({\varLambda }\) with degree less than or equal to N

\({}_{x}^{}D^{\mu }_{1}\) :

Right Riemann–Liouville fractional derivative operator with \(\mu >0\)

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Acknowledgements

The authors are greatly indebted to the referees for useful comments. This work was supported by the National Natural Science Foundation of China (11771112, 11671112).

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Authors and Affiliations

  1. School of Mathematical Science, Qufu Normal University, Qufu, 273165, China

    Jingjun Zhao

  2. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Yanming Zhang & Yang Xu

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  1. Jingjun Zhao
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  2. Yanming Zhang
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  3. Yang Xu
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Correspondence to Yang Xu.

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Communicated by José Tenreiro Machado.

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Zhao, J., Zhang, Y. & Xu, Y. Implicit Runge–Kutta and spectral Galerkin methods for Riesz space fractional/distributed-order diffusion equation. Comp. Appl. Math. 39, 47 (2020). https://doi.org/10.1007/s40314-020-1102-3

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