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A Comparative Study of Numercial Solution of PDE in Geosciences
Author(s) -
Hongren Zhang
Publication year - 1994
Publication title -
acta geologica sinica ‐ english edition
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.444
H-Index - 61
eISSN - 1755-6724
pISSN - 1000-9515
DOI - 10.1111/j.1755-6724.1994.mp7001007.x
Subject(s) - finite element method , thermal conduction , partial differential equation , mathematics , laplace transform , finite difference , finite difference method , laplace's equation , mixed finite element method , algebraic equation , groundwater flow equation , algebraic number , heat equation , mathematical analysis , groundwater flow , groundwater , physics , thermodynamics , geology , nonlinear system , aquifer , geotechnical engineering , quantum mechanics
A number of phenomena and processes in geosciences can be summarized by second order partial differential equations. The major numerical methods for their solution include the classical finite difference method and the finite element method newly developed in the last two or three decades. Since 1977 the author has proved that for the Laplace and Poisson equations, these two methods are identical and are different only in the process of formulation. For transient problems, such as heat conduction in the earth and the groundwater and oil‐gas unsteady flow in porous media, there are some differences in resulting linear algebraic euqations. In general, two methods give similar results, but when the time step is decreased to some extent, the resulting. algebraic equation will be consistent with the anti‐heat conduction equation rather than the original heat conduction equation. This is the reason why unrealistic potentials are produced by the finite element method. Such a problem can be overcome by using the lumped mass procedure, but it makes the two methods identical again. To improve the traditional finite difference method, it is quite desirable to introduce the common practice of the finite element method to define the parameters in elements rather than on nodes.