# Introduction to Numerical Modelling A mathematical model typically consists of one equation or several coupled equations. To simulate the system described by the model, these equations have to be solved.

An analytic solution is a function or relation that describes the state of the system and depends on model parameters. In most cases, it is extremely difficult to find analytical solutions of a given mathematical model.

To obtain an approximate solution, the model can be solved numerically. The numerical solution of the model equations consists of values (numbers) that describe the state of the system.

Many dynamical processes occurring in nature can be described by differential equations together with a set of suitable boundary conditions. To solve differential equations numerically, the integration domain is discretized into a finite number of integration points also called numerical grid or mesh (see figure above).

In Finite Difference (FD) methods, differential operators are approximated by a discrete expression. The spacing between the discrete elements determines the accuracy of the solution and the convergence rate of the integration scheme.

Finite Element (FE) methods are predominantly used to solve problems where the integration domain is of irregular shape. FE methods use a variational approach to solve differential equations numerically.