Page 18 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

4 1 Introduction

translated into a mathematical model, which is usually comprised of a set of partial
differential equations. Due to the complex and complicated nature of these equa-
tions, it is very difficult, if not impossible, to find analytical solutions. Alternatively,
numerical simulation methods need to be used to find approximate solutions for the
problem.
To ensure the accuracy and reliability of the numerical simulation solution, it is
necessary to investigate the solution characteristics of the partial differential equa-
tions through a theoretical analysis. For example, some theoretical methods can be
used to investigate the solution singularity and multiple solution characteristics of
the partial differential equations, as well as the conditions under which such char-
acteristics can occur. If possible, a benchmark model should be established for a
particular kind of geoscience problem. The geometrical nature and boundary con-
ditions of this benchmark model can be further simplified, so that the theoretical
solution, known as the benchmark solution, can be obtained. This benchmark solu-
tion is valuable and indispensable for the verification of both the numerical algo-
rithm and the computer code, which are used to solve the problem that is generally
characterised by a complicated geometrical shape and complex material properties.
It must be pointed out that, due to the approximate nature of a numerical method,
the theoretical investigation of the solution characteristics associated with the partial
differential equations of a problem plays an important role in applying the research
methodology of computational geoscience to solve real problems. This is the key
step to ensure the accuracy and reliability of the numerical solution obtained from
the numerical simulation of the problem.

1.2.3 The Numerical Simulation Model of a Geoscience Problem

From a mathematical point of view, the numerical simulation model of a geoscience
problem can be also called the discretized type of mathematical model. Both the
finite element method and the finite difference method are commonly-used dis-
cretization methods for numerical simulation of geoscience problems (Zienkiewicz
1977, Zhao et al. 1998a, 2006a). The basic idea behind these numerical methods is
to translate the partial differential equations used to describe the geoscience problem
in a continuum system, into the corresponding algebraic equations in a discretized
system, which in turn is usually comprised of a large number of elements. Through
solving the resulting algebraic equations of the discretized system, a numerical solu-
tion can be obtained for the problem.
Compared with an engineering problem, a geoscience problem commonly has
both large length-scale and large time-scale characteristics. The length-scale of a
geoscience problem is commonly measured in either tens of kilometers or even
hundreds of kilometers, while the time-scale of a typical problem is often measured
in several million years or even several tens of million years. In addition, most geo-
science problems are coupled across both multiple processes and multiple scales.
Due to these significant differences between engineering problems and geoscience
problems, commercial computer programs and related algorithms, which are mainly