Discretization Methods-Mathematical Modeling and Simulation-Lecture Slides, Slides of Mathematical Modeling and Simulation

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Lecture Slides

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Modeling and Simulation

Lecture One: Introduction to Modeling

Descritization Methods

• Main descritization approaches include
  • finite differences,
  • finite volumes and
  • finite elements.
• Although, all of these methods result in linear systems of

similar structure, the desired characteristics of meshes

for these methods differ.

• In general, for all types of meshes, there are certain

properties to be controlled.

• These are local density of points, smoothness of the

point distribution and shape of grid volumes.

The finite difference method

• It is one of the techniques used to obtain an approximate solution to ordinary and partial differential equation based models governing the behavior of physical systems by using neighboring points.
• It imposes a regular grid on the physical domain, and then approximates the derivative of unknown functions at a grid point by the ratio of the difference in the function value at two adjacent points to the distance between the grid points.
• Although irregular grids can be used for finite difference schemes, regular grids are mostly employed with simple difference schemes.
• However, there can be serious problems with the finite difference scheme at coordinate singularities.

Finite volume method

• In finite volume method, the physical space is divided into

small volumes and the ordinary or partial differentials

are integrated over each of these volumes.

• Then, the variables are approximated by their average

values in each volume and the changes through the

surfaces of each volume are approximated as a function

of the variables in neighboring volumes.

• The finite volume descritization can be used for both

regular and irregular meshes.

• The changes through the surfaces are also well-defined

in an irregular mesh.

Grids in Finite Difference Approach

• The order of a finite-difference approach is given by a Taylor series expansion for the numerical formulation. For instance, the first derivative a function f is given by a three point (equally spaced) formula that ensures a second order accuracy.
• This means that if resolution is doubled, accuracy increases by a factor of 4.
• Higher order formulations are possible but most finite difference approaches use second order schemes.
• In more than one-dimensional problem, the best accuracy is obtained by using Cartesian-like grids and one may include includes curvilinear grids for even better shapes.

Grids in Finite Difference Approach

• When the orientation of the boundary does not correspond to that of the grid, descritization of the boundary introduces a series of artificial jumps along the boundary as shown in Fig. 1.
• Curvilinear models exist that tend to follow the boundaries, but they usually fail as soon as the complexity of the system grows with many concave and convex changes.
• Another example is the description of straits when only few points are available (Fig. 2). In that situation, the strait width must take values in a set of discrete numbers at the price of misrepresenting the width and therefore will have errors in the exchange of energy, momentum and flow rates.

Grids in Finite Difference Approach

Figure 2 The uneven boundaries and square lattice representation.

Model^ Real Boundary boundary

Grids in Finite Difference Approach

• Several standard staggering techniques are used in finite

difference modeling. These include the non-staggered A-

type, B-type and C-type grids respectively as illustrated

in Fig. 3.

A-type (^) B-type C-type

Fig. 3 Three different types of square grid formulations.