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### Time and space discretization

In the real world, the space and time are continuous, but in an MHD simulation, we have to use discrete space and time to describe the system.

The reasons why we can not use a continuous description of the system are as follows:

1.
In order to describe the system continuously in space, we have to save all the information at every point of the system at specific times. This will unavoidably lead to excessive use of computer resources, such as memory, hard disk, computation speed. In order to describe the system continuously in time, we have to have an infinitesimal time step. Thus we will never be able to run the simulation even for a very short time because of the computation limitations.
2.
In MHD theory, the variables, such as density, pressure, resistivity and other quantities are integrated values over a certain region. So it is sufficient to use a value to describe it in the region. Thus it is unnecessary to describe these variables in a infinitesimal space.
3.
The practical application of discrete time and space in MHD simulations has proved that, as long as we obey certain rules in our simulation to determine the time step and space step, we can accurately describe the system evolution.

Since we have to adopt the discrete description of the environment, we have to consider how to make time and space discrete. The simplest way to discretize space is to divide the whole simulation region into many regions of the same size. For the time discretion, the simplest way is to choose the same time step, dt, for all the evolution. The uniform discretization is shown in Figure 1.

 Figure 1: One-dimension and two-dimension uniform-space discretization and time-uniform discretization.

Though the uniform discretization of space and time is simple, we can not hope that it can optimize the simulation efficiency and reliability. Figure 2 shows a typical physical component in simulation plot in one dimension at a given time. From the figure, we can see that in interval ab, the slope of the variable is steep. While in interval cd, the plot is quite flat. If we still use the uniform step discretion, we will be unable to have a better representation of the real process in ab, as numerical errors may heavily deviate the real phenomenon. While at the same time, more than enough grids will be used in cd, which will unnecessarily cost more computation resources. In order to have a better solution of this problem, we need finer grids in region ab, while at the same time, coarser grids are enough in region cd. A gridding scheme depending on the real physical component value, instead of being predefined, is called adaptive. The kind of step adjusting that can be done by program itself is called self-adaptive. The same thing can also be performed in time domain.

 Figure 2: A simple case where it is better to use adaptive discretization.

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