The solution of two-dimensional advection-diffusion equations via operational matrices

详细信息    查看全文

• 作者：Francisco de la Hoz ; Fern ; o Vadillo
• 关键词：Advection&ndash ; diffusion equations ; Pseudo-spectral methods ; Chebyshev differentiation matrices ; Hermite differentiation matrices ; Matrix exponentials ; Sylvester equations
• 刊名：Applied Numerical Mathematics
• 出版年：2013
• 出版时间：October, 2013
• 年：2013
• 卷：72
• 期：Complete
• 页码：172-187
• 全文大小：621 K

文摘

In this paper we describe a spectrally accurate, unconditionally stable, efficient method using operational matrices to solve numerically two-dimensional advection-diffusion equations on a rectangular domain.

The novelty of this paper is to relate for the first time evolution partial differential equations and Sylvester-type equations, avoiding Kronecker tensor products. Furthermore, to reach large times, the calculation of just two matrix exponentials is required, for which we compare different techniques based on Pad¨¦?s approximations, matrix decompositions and Krylov spaces, as well as a new technique which avoids the computation of matrix exponentials. We also illustrate how to take advantage of multiple precision arithmetic.

Finally, possible generalizations to non-linear problems and higher-dimensional problems, as well as to unbounded domains, are considered.