# Numerical simulation

Computer simulations that are carried out using numerical methods such as turbulence models are generally referred to as **numerical simulation** . Well-known examples are weather and climate forecasts , numerical flow simulations or strength and rigidity calculations.

## method

Numerical simulations can be divided into the following steps:

### Modeling

In the modeling (model structure) the basic properties of a simulation are formulated in the form of mathematical models. The models are usually developed independently of a specific task.

### Parameterization

During the parameterization, models are selected, equipped with specific calculated values and linked with one another in such a way that the overall model represents a specific application as well as possible. Inaccurate knowledge of the models or the boundary conditions is the most common source of errors in simulations.

### calculation

The numerical methods are special calculation procedures that fall under the sub-area of numerical mathematics . The actual calculation is done by starting a solution program, the so-called *solver* . This carries out the actual calculation and saves the calculation results. Since a closed solution of the systems is usually not possible, iterative solution methods are used to find an approximate solution. Almost all simulation calculations require very large amounts of data to be processed. Nevertheless, the computing time can vary greatly depending on the simulation method. For this reason, parallel computers , vector computers or PC clusters are often used in this area , with many individual computers working on a result at the same time. However, the speed of such calculations cannot be increased at will, as the number of computing cores involved generally increases the communication effort ( scalability ).

### Evaluation and presentation

The results of the calculation are called raw data. These are available as digital result files that now have to be processed in such a way that they are understandable for people. The evaluation required for this is an elementary part of the simulation. On the one hand, statistical methods are used for the evaluation, which summarize or analyze data. An important aspect, however, is also the ability to process data graphically.

## Areas of application

The mathematical problems of numerical simulations can often be traced back to the solution of differential equations, the solution of eigenvalue and eigenvector problems, the solution of linear systems of equations or the calculation of integrals. Due to the complexity of the simulation programs and the uncertainty of the applied parameters and boundary conditions, accompanying procedures, such as analytical calculations, are often used in parallel to check the results.

The complexity of different numerical simulations is very different. Therefore, problems such as strength calculations or vibration analyzes of buildings and machine parts have become standard tools for designers - in other processes (weather forecasts, climate calculations), on the other hand, one moves at or beyond the limits of the capabilities of modern computers. There are also fundamental problems such as the chaotic behavior of many dynamic systems.

Numerical simulations can be used in many different ways. Some important or well-known examples are:

### Natural sciences

- Biology: bacterial reproduction, mutations, genetic engineering , ecology, biodiversity
- Theoretical chemistry
- Physics: standard model of elementary particles, lattice theory , phase transitions , fluid mechanics
- Astronomy: Big Bang , Nuclear Fusion
- Meteorology: weather and climate models
- Geosciences: plate tectonics , tides , earthquakes , mountain formation , hydraulic and hydrochemical processes

### Engineering

- Architecture and civil engineering: Static and dynamic strength calculations (buildings, bridges)
- Chemical and process engineering: Combustion processes and chemical reactions (internal combustion engines, yield from chemical syntheses)
- Mechanical engineering: flight simulators, vibration analysis on electrical machines , stresses and deformations (elastic and plastic, e.g. virtual crash tests using finite element methods )
- Technical physics: semiconductor components, heat conduction processes, optical systems (lens systems, lasers, thermal deformations due to absorption), fusion reactors , accelerators and nuclear reactions

### Economics

- Operations Research Problems

### military

- Nuclear weapons , explosives, propellants and projectiles

### entertainment

- Computer games (consideration of physical properties and lighting)

## Examples

One area in which numerical simulations are used is flow simulations. Air flows are determined by a computer model, the space of which is divided into a grid consisting of cells or voxels ( discretization ).

The process is somewhat similar to the digital representation of photos on the computer, which now consist of individual image points ( pixels ). Each pixel has only a single color value, although the real image is actually continuous, i.e. In other words, areas are combined into areas of the same color. If the viewing distance is sufficiently large, the color values seem to flow together again to form a continuous image for the eye. If the resolution of the digital image display is too low, the photo appears blurred or step-like.

Unlike a pixel image, which has only two spatial dimensions and one color information, flow simulations normally consist of three spatial dimensions. For each of the points there are - depending on the problem - several parameters, which in turn can be dependent on one another. The physical quantities (e.g. pressure or temperature) of neighboring grid points change in the course of the calculation due to mutual influences.

In numerical simulation on a grid, the rules for resolution are similar to those for displaying photos on a computer. If the spatial resolution is too low (large cells), then the physics is not mapped well and inaccuracies occur. Therefore, one is interested in the highest possible spatial resolution. On the other hand, with a high resolution, the computing power is often insufficient to obtain a result in an acceptable time. For example, dividing it into 100 × 100 × 100 cells results in one million points. If you halve the edge length of these cells, the number increases to eight million. Even with modern computers, the resolution therefore quickly reaches the limits of computing power.

Simulations in other areas of application use systems that not only consist of three spatial dimensions but, for example, of three spatial and one temporal dimensions. There can also be a large number of parameters for each of the grid points. In addition to the cubic grid shape described, which often results from the discretization of the dimensions, other grid shapes are also used for the simulation, for example with the finite element method. There are also simulations that do not use a lattice structure; particle systems such as the simple model of hard spheres are an example of this.