# Numerical methods

*Kimmo Hentinen, Geological Survey of Finland, P.O. Box 1237, FI-70211 KUOPIO, FINLAND, kimmo.hentinen[at]gtk.fi*

## Introduction

Numerical methods stands for numerical methods for solving differential or partial differential equations. Mathematical modelling leads often to partial differential equations (PDE). In ‘realistic’ models these equations are very hard or impossible to solve (Wang & Anderson 1982). Numerical methods provide a way to achieve approximative solution for these equations. Most common methods are finite difference method (FDM), finite element method (FEM) and finite volume method (FVM). There are also some other methods closely related to ones mentioned before (Diersch 2014).

In each of these numerical methods computational domain is divided into finite number of elements or nodes. The solution of the governing PDE is estimated in these discrete elements or points depending on the method in use. This article will describe the ideas behind FDM and FEM, but won’t go into details of these methods. These methods are selected because they are probably most widely used. FVM is left out because it is closely related to FEM and has many similar properties to FEM (Diercsh 2014).

## Description of the methods

Finite difference method (FDM) is the classic numerical approach and it was the first widespread numerical method e.g. in groundwater modelling. In FDM the computational domain is covered with rectangular grid and using this grid the derivates in partial differential equations are estimated (Kovarik 2000). Estimation is done using the actual values in the nodes of the grid. This procedure generates a system of linear equations which can be written in matrix form. This method requires an initial guess from where the approximative solution is processed forward using iterative methods until we have a solution for the matrix equation which is accurate enough. (Wang & Anderson 1982).

In finite element method (FEM) the computational domain is divided into finite number of elements. These elements can be e.g. in triangular shape. For each of the elements solution of the governing PDE is estimated using certain basic functions and coefficients for each function. This estimative solution is inserted into the governing PDE and with some modifications to the governing equations the coefficients can be solved from a system of linear equations (Hämäläinen & Järvinen 2006). This is transformed in to a matrix equation which is again solved with iterative methods. Approximative solution is achieved from the basic functions and their coefficients. This is the normal procedure but FEM has to be ‘tweaked’ a bit according to the governing equations (Hämäläinen & Järvinen 2006). In FVM the idea is the same but the process from the governing equations to the matrix equation is different from FEM.

## Appropriate applications

Finite difference method has some restrictions compared to FEM. Grid used in FDM has to be rectangular and thus domain boundary can’t be matched exactly. This is not a problem in groundwater flow because the boundary itself isn’t known exactly anyway (Kovarik 2000). Rectangular grid generates also another problem. Local refinement of the grid generates refinement also in areas where it isn’t necessary and thus increasing computational cost. In FEM triangular grid can be unstructured and thus domain boundary has more realistic shape and local refinement can be done more effectively.

Finite difference method is also restricted to simple boundary condition where FEM can handle quite arbitrary boundary conditions (Diercsh 2014). Mathematically and computationally FEM is heavier than quite simple FDM, but it is more versatile. With modern computational capacity FEM being heavier is usually not a problem. Mathematical models tend to be more accurate and FEM is considered to be most general and powerful compared with other numerical methods (Diercsh 2014).

## References

Kovarik, K. 2000, Numerical Models in Groundwater Pollution. 221 pages.

Wang, H.F. & Anderson, M.P. 1982. Introduction to groundwater modeling, Academic Press. 237 pages.

Diersch, H-J.G. 2014, FEFLOW Finite Element Modeling of Flow, Mass and Heat Transport in Porous and Fractured Media, Springer. 996 pages.

Hämäläinen, J. & Järvinen, J. 2006. Elementtimenetelmä virtauslaskennassa, CSC – Tieteellinen laskenta Oy. 215 pages.

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