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Volume 9, Issue 1
An Optimal-Order Error Estimate for a Finite Difference Method to Transient Degenerate Advection-Diffusion Equations

T. Lu & J. Jia

Int. J. Numer. Anal. Mod., 9 (2012), pp. 56-72.

Published online: 2012-09

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  • Abstract

We prove an optimal-order error estimate in a degenerate-diffusion weighted energy norm for implicit Euler and Crank-Nicolson finite difference methods to two-dimensional time-dependent advection-diffusion equations with degenerate diffusion. In the estimate, the generic constants depend only on certain Sobolev norms of the true solution but not on the lower bound of the diffusion. This estimate, combined with a known stability estimate of the true solution of the governing partial differential equations, yields an optimal-order estimate of the finite difference methods, in which the generic constants depend only on the Sobolev norms of the initial and right-hand side data.

  • Keywords

Convergence analysis, degenerate advection-diffusion equations, finite difference methods, optimal-order error estimates.

  • AMS Subject Headings

65N06, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-9-56, author = {T. Lu and J. Jia}, title = {An Optimal-Order Error Estimate for a Finite Difference Method to Transient Degenerate Advection-Diffusion Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2012}, volume = {9}, number = {1}, pages = {56--72}, abstract = {

We prove an optimal-order error estimate in a degenerate-diffusion weighted energy norm for implicit Euler and Crank-Nicolson finite difference methods to two-dimensional time-dependent advection-diffusion equations with degenerate diffusion. In the estimate, the generic constants depend only on certain Sobolev norms of the true solution but not on the lower bound of the diffusion. This estimate, combined with a known stability estimate of the true solution of the governing partial differential equations, yields an optimal-order estimate of the finite difference methods, in which the generic constants depend only on the Sobolev norms of the initial and right-hand side data.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/611.html} }
TY - JOUR T1 - An Optimal-Order Error Estimate for a Finite Difference Method to Transient Degenerate Advection-Diffusion Equations AU - T. Lu & J. Jia JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 56 EP - 72 PY - 2012 DA - 2012/09 SN - 9 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/611.html KW - Convergence analysis, degenerate advection-diffusion equations, finite difference methods, optimal-order error estimates. AB -

We prove an optimal-order error estimate in a degenerate-diffusion weighted energy norm for implicit Euler and Crank-Nicolson finite difference methods to two-dimensional time-dependent advection-diffusion equations with degenerate diffusion. In the estimate, the generic constants depend only on certain Sobolev norms of the true solution but not on the lower bound of the diffusion. This estimate, combined with a known stability estimate of the true solution of the governing partial differential equations, yields an optimal-order estimate of the finite difference methods, in which the generic constants depend only on the Sobolev norms of the initial and right-hand side data.

T. Lu and J. Jia. (2012). An Optimal-Order Error Estimate for a Finite Difference Method to Transient Degenerate Advection-Diffusion Equations. International Journal of Numerical Analysis and Modeling. 9 (1). 56-72. doi:
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