Volume 16, Issue 2
A Second-Order Crank-Nicolson Method for Time-Fractional PDEs

Int. J. Numer. Anal. Mod., 16 (2019), pp. 225-239.

Published online: 2018-10

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• Abstract

Based on convolution quadrature in time and continuous piecewise linear finite element approximation in space, a Crank-Nicolson type method is proposed for solving a partial differential equation involving a fractional time derivative. The method achieves second-order convergence in time without being corrected at the initial steps. Optimal-order error estimates are derived under regularity assumptions on the source and initial data but without having to assume regularity of the solution.

• Keywords

Crank-Nicolson scheme, time-fractional equation, convolution quadrature, finite element method, error estimates.

35R11, 26A33, 33F05

mgunzburger@fsu.edu (Max Gunzburger)

jwang@math.msstate.edu (Jilu Wang)

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@Article{IJNAM-16-225, author = {Max and Gunzburger and mgunzburger@fsu.edu and 11362 and Department of Scientific Computing, Florida State University, Tallahassee, FL 32304, USA and Max Gunzburger and Jilu and Wang and jwang@math.msstate.edu and 11916 and Department of Mathematics and Statistics, Mississippi State University, Starkville, MS 39762, USA and Jilu Wang}, title = {A Second-Order Crank-Nicolson Method for Time-Fractional PDEs}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {16}, number = {2}, pages = {225--239}, abstract = {

Based on convolution quadrature in time and continuous piecewise linear finite element approximation in space, a Crank-Nicolson type method is proposed for solving a partial differential equation involving a fractional time derivative. The method achieves second-order convergence in time without being corrected at the initial steps. Optimal-order error estimates are derived under regularity assumptions on the source and initial data but without having to assume regularity of the solution.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/12801.html} }
TY - JOUR T1 - A Second-Order Crank-Nicolson Method for Time-Fractional PDEs AU - Gunzburger , Max AU - Wang , Jilu JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 225 EP - 239 PY - 2018 DA - 2018/10 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/12801.html KW - Crank-Nicolson scheme, time-fractional equation, convolution quadrature, finite element method, error estimates. AB -

Based on convolution quadrature in time and continuous piecewise linear finite element approximation in space, a Crank-Nicolson type method is proposed for solving a partial differential equation involving a fractional time derivative. The method achieves second-order convergence in time without being corrected at the initial steps. Optimal-order error estimates are derived under regularity assumptions on the source and initial data but without having to assume regularity of the solution.

Max Gunzburger & Jilu Wang. (2020). A Second-Order Crank-Nicolson Method for Time-Fractional PDEs. International Journal of Numerical Analysis and Modeling. 16 (2). 225-239. doi:
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