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Volume 20, Issue 6
Unconditionally Energy Stable and First-Order Accurate Numerical Schemes for the Heat Equation with Uncertain Temperature-Dependent Conductivity

Joseph Anthony Fiordilino & Matthew Winger

DOI: 10.4208/ijnam2023-1035

Int. J. Numer. Anal. Mod., 20 (2023), pp. 805-831.

Published online: 2023-11

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  • Abstract

In this paper, we present first-order accurate numerical methods for solution of the heat equation with uncertain temperature-dependent thermal conductivity. Each algorithm yields a shared coefficient matrix for the ensemble set improving computational efficiency. Both mixed and Robin-type boundary conditions are treated. In contrast with alternative, related methodologies, stability and convergence are unconditional. In particular, we prove unconditional, energy stability and optimal-order error estimates. A battery of numerical tests are presented to illustrate both the theory and application of these algorithms.

  • Keywords

Time-stepping, finite element method, heat equation, temperature-dependent thermal conductivity, uncertainty quantification.

  • AMS Subject Headings

65M12, 65M15, 65M60, 80A20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-20-805, author = {Fiordilino , Joseph Anthony and Winger , Matthew}, title = {Unconditionally Energy Stable and First-Order Accurate Numerical Schemes for the Heat Equation with Uncertain Temperature-Dependent Conductivity}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2023}, volume = {20}, number = {6}, pages = {805--831}, abstract = {

In this paper, we present first-order accurate numerical methods for solution of the heat equation with uncertain temperature-dependent thermal conductivity. Each algorithm yields a shared coefficient matrix for the ensemble set improving computational efficiency. Both mixed and Robin-type boundary conditions are treated. In contrast with alternative, related methodologies, stability and convergence are unconditional. In particular, we prove unconditional, energy stability and optimal-order error estimates. A battery of numerical tests are presented to illustrate both the theory and application of these algorithms.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2023-1035}, url = {http://global-sci.org/intro/article_detail/ijnam/22142.html} }
TY - JOUR T1 - Unconditionally Energy Stable and First-Order Accurate Numerical Schemes for the Heat Equation with Uncertain Temperature-Dependent Conductivity AU - Fiordilino , Joseph Anthony AU - Winger , Matthew JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 805 EP - 831 PY - 2023 DA - 2023/11 SN - 20 DO - http://doi.org/10.4208/ijnam2023-1035 UR - https://global-sci.org/intro/article_detail/ijnam/22142.html KW - Time-stepping, finite element method, heat equation, temperature-dependent thermal conductivity, uncertainty quantification. AB -

In this paper, we present first-order accurate numerical methods for solution of the heat equation with uncertain temperature-dependent thermal conductivity. Each algorithm yields a shared coefficient matrix for the ensemble set improving computational efficiency. Both mixed and Robin-type boundary conditions are treated. In contrast with alternative, related methodologies, stability and convergence are unconditional. In particular, we prove unconditional, energy stability and optimal-order error estimates. A battery of numerical tests are presented to illustrate both the theory and application of these algorithms.

Joseph Anthony Fiordilino & Matthew Winger. (2023). Unconditionally Energy Stable and First-Order Accurate Numerical Schemes for the Heat Equation with Uncertain Temperature-Dependent Conductivity. International Journal of Numerical Analysis and Modeling. 20 (6). 805-831. doi:10.4208/ijnam2023-1035
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