Kouli_etal_2008_Groundwater modelling_BOOK.pdf - Pantelis ...

Analytical and Numerical Solutions... 287
equation and we can generalize the solutions to arbitrary initial conditions. The decomposition
methods allow a reduction of the computational complexity and their higher-order
schemes achieved the forced accuracy.
We could confirm also the new methods with the analytical and numerical test examples
and present the higher-order results of the underlying schemes.
The problem for the convection-dominant equation can be solved with combined analytical
and decomposed methods to decouple the complicated equation systems and achieve
the accuracy with iterative or analytical embedded methods.
For complex computations of such convection-dominant problems, we use these methods
in the initialization process of the computation and switch after sufficient accuracy to
implicit methods with large time-steps.
In future the decomposition methods and analytically-improved methods can be generalized
for non-smooth and non-linear problems in time and space.
References
[1] M. Abramowitz, I.A. Stegun. Handbook of Mathematical Functions. Dover Publications
New York, 1970.
[2] P. Bastian, K. Birken, K. Eckstein, K. Johannsen, S. Lang, N. Neuss, and H. Rentz-
Reichert. UG - a flexible software toolbox for solving partial differential equations.
Computing and Visualization in Science, 1(1):27–40, 1997.
[3] H. Bateman. The solution of a system of differential equations occurring in the theory
of radioactive transformations. Proc. Cambridge Philos. Soc., v. 15, pt V:423–427,
1910.
[4] T. Barth and M. Ohlberger. Finite volume methods: foundation and analysis. In: E.
Stein, R. de Borst, T.J.R. Hughes (Eds.). Encyclopedia of Computational Mechanics,
John Wiley & Sons, Ltd, 2004.
[5] M.A. Celia, T.F. Russell, I. Herrera, and R.E. Ewing. An Eulerian-Lagrangian localized
adjoint method for the advection-diusion equation. Advances in Water Resources,
13, 187-206, 1990.
[6] D. Daoud, J. Geiser. Overlapping Schwarz Wave-Form Relaxation for the Solution
of Coupled and Decoupled Systems of Convection Diffusion Reaction Equation. Applied
Mathematics and Computations, Elsevier, North Holland, 190 (1), 946–964,
2007.
[7] C. N. Dawson, Q. Du, and D. F. Dupont. A finite Difference Domain Decomposition
Algorithm for Numerical solution of the Heat Equation. Mathematics of Computation,
57, 63-71, 1991.
[8] B. Davis. Integral Transform and Their Applications. Applied Mathematical Sciences,
25, Springer Verlag, New York, Heidelberg, Berlin, 1978.