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Volume 11, Issue 1
An $hp$-Version of $C^0$ -Continuous Petrov-Galerkin Time-Stepping Method for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels

Shuangshuang Li, Lina Wang & Lijun Yi

East Asian J. Appl. Math., 11 (2021), pp. 20-42.

Published online: 2020-11

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  • Abstract

An $hp$-version of $C^0$-CPG time-stepping method for second-order Volterra integro-differential equations with weakly singular kernels is studied. In contrast to the methods reducing second-order problems to first-order systems, here the CG and DG methodologies are combined to directly discretise the second-order derivative. An a priori error estimate in the $H^1$-norm, fully explicit with respect to the local discretisation and regularity parameters, is derived. It is shown that for analytic solutions with start-up singularities, exponential rates of convergence can be achieved by using geometrically refined time steps and linearly increasing approximation orders. Theoretical results are illustrated by numerical examples.

  • AMS Subject Headings

65R20, 65M60, 65M15

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COPYRIGHT: © Global Science Press

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@Article{EAJAM-11-20, author = {Li , ShuangshuangWang , Lina and Yi , Lijun}, title = {An $hp$-Version of $C^0$ -Continuous Petrov-Galerkin Time-Stepping Method for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels}, journal = {East Asian Journal on Applied Mathematics}, year = {2020}, volume = {11}, number = {1}, pages = {20--42}, abstract = {

An $hp$-version of $C^0$-CPG time-stepping method for second-order Volterra integro-differential equations with weakly singular kernels is studied. In contrast to the methods reducing second-order problems to first-order systems, here the CG and DG methodologies are combined to directly discretise the second-order derivative. An a priori error estimate in the $H^1$-norm, fully explicit with respect to the local discretisation and regularity parameters, is derived. It is shown that for analytic solutions with start-up singularities, exponential rates of convergence can be achieved by using geometrically refined time steps and linearly increasing approximation orders. Theoretical results are illustrated by numerical examples.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.020520.120620}, url = {http://global-sci.org/intro/article_detail/eajam/18411.html} }
TY - JOUR T1 - An $hp$-Version of $C^0$ -Continuous Petrov-Galerkin Time-Stepping Method for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels AU - Li , Shuangshuang AU - Wang , Lina AU - Yi , Lijun JO - East Asian Journal on Applied Mathematics VL - 1 SP - 20 EP - 42 PY - 2020 DA - 2020/11 SN - 11 DO - http://doi.org/10.4208/eajam.020520.120620 UR - https://global-sci.org/intro/article_detail/eajam/18411.html KW - $hp$-version, second-order Volterra integro-differential equation, weakly singular kernel, continuous Petrov-Galerkin method, exponential convergence. AB -

An $hp$-version of $C^0$-CPG time-stepping method for second-order Volterra integro-differential equations with weakly singular kernels is studied. In contrast to the methods reducing second-order problems to first-order systems, here the CG and DG methodologies are combined to directly discretise the second-order derivative. An a priori error estimate in the $H^1$-norm, fully explicit with respect to the local discretisation and regularity parameters, is derived. It is shown that for analytic solutions with start-up singularities, exponential rates of convergence can be achieved by using geometrically refined time steps and linearly increasing approximation orders. Theoretical results are illustrated by numerical examples.

Shuangshuang Li, Lina Wang & Lijun Yi. (2020). An $hp$-Version of $C^0$ -Continuous Petrov-Galerkin Time-Stepping Method for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels. East Asian Journal on Applied Mathematics. 11 (1). 20-42. doi:10.4208/eajam.020520.120620
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