MPIAMRVAC
2.2
The MPI  Adaptive Mesh Refinement  Versatile Advection Code

Numerical methods for conservation laws can in general be divided into two approaches: a onestep approach, where time and space discretization is coupled and semidiscrete schemes, where time and space integration are separated. AMRVAC chooses to use the latter approach, and provides multiple time discretization methods. They can all be found in mod_advance.t. Please not that not all the schemes could be combined with provided time discretizaiton methods.
If we write a conservation equation into the semidiscrete form, like,
Then, PDE becomes ODE, and if we discrete into:
We will have,
This is the famous onestep Forward Euler method, and called onestep in AMRVAC.
In this method, deltax is decided by the mesh, while deltat is constrained by the CFL condition:
S_max is the maximum wave velocity at the current moment and the small c is called CFL number, and the onestep Euler method requires the CFL number to be between 0 and 1. As a user, we want the CFL number to be as large as possible, namely, 1. However, since the maximum velocity estimated in AMRVAC would be lower than the actual value, so that the CFL number should be put only to 0.8 or 0.9 in real simulations, usually. But in the following discriptions, we will still use the theoretic value, namely, 1.
Then, since most of the spatial discretization in AMRVAC are secondorder, we need at least a secondorder temporal discretization. Two method provided in AMRVAC is the famous midpoint method (time_stepper='twostep' and time_integrator='Predictor_Corrector'):
and the Heun's (time_integrator='ssprk2') method:
Actually, they are equivalent to the Midpoint Riemann sum and Trapezoidel rule in numerical integral methods, respectively.
Then, in AMRVAC, we have higher order time discretization methods. But before introducing them, we should know that either firstorder or secondorder methods can be recognised as firstorder or secondorder RungeKutta method, respectively. And to be more generally, we can get mth order RK method:
where
This is the form you can usually see in text books. However, in more recent literatures, this equation will be rewriten in another equivalent form:
For example, the classic RK4 written in this form would be:
This is exactly what we call rk4 in AMRVAC.
Besides rk4, AMRVAC also provide other fourthorder RK methods, namely, jameson (variant by Antony Jameson). They are simpler than the classic one, for example, the Jameson variant is:
Due to their simplicity, they are used by some other codes like MURaM instead of classic RK4.
As we know, with different combination of coefficients, we can have as many varients of RK4 as we want. Then, what is the difference between these methods? Actually, these methods can perform similarly in linear problems. However, in nonlinear problems, the classic RK4 performs to be a true fourthorder method.
Nevertheless, even the classic RK4 suffers from some drawbacks, namely, it cannot fulfill the TVD condition. As we know, most TVD schemes are derived with onestep Euler method. In the spatial discretization, firstorder upwind scheme can always fulfill the TVD condition, but higherorder schemes may not. Similarly, the onestep Euler method can always fulfill the TVD condition, but higherorder RK methods may not. Thus, we shall also pursue a methond that fulfill the TVD condition for higherorder RK methods, especially when using a TVD scheme. Actually, for a RK method, if all the coefficients alpha_{ik} and beta_{ik} are not negative, if the CFL number can fulfill:
This RK method would fulfill the TVD condition. AMRVAC provides the thirdorder TVD RK method(time_stepper='threestep' and time_integrator='ssprk3'):
This thirdorder RK method, a.k.a. ShuOsher method, can allow the CFL number to be 1, the same with the onestep Euler method. Therefore, it is favoured by many users.
Several years after the ShuOsher method is proposed, the research of RK methods moved from TVD condition to the socalled strong stability preserving or SSP condition. Actually these two concepts are similar, but the SSP might have broader meaning. Anyway, we will still use the term TVD in the following discriptions.
For the thirdorder TVD RK method, the CFL number could be as large as 1. While the cost is three steps. Generally, for an mstep sorder RK method, there must be m >= s. So, if we want m to be as small as possible, we should have m = s. However, it is proved that if we want an sstep sorder RK method to fulfill the TVD conditon, s could not be more than 3. Otherwise, the function g(t,u) should be reversible. However, even g(t,u) is reversible, considering that double time cost would be needed, we seldom consider an sstep sorder RK method with s more than 3. At the same time, it is also proved that, the CFL number of an mstep sorder RK method could not be more than ms+1, which means that for a fourstep thirdorder RK method, theoretically we can allow the CFL number to be 2, which seems to be costeffective. The following is this fourstep thirdorder RK method in AMRVAC, or time_stepper='fourstep' with time_integrator='ssprk4':
But note that this upper limit ms+1 could not be met by most RK methods, for example, for the fivestep fourthorder RK method, the CFL number could only be 1.508 instead of 2. Anyway, even with 1.508, the effective CFL number (means CFL per step) is larger than ShuOsher method. Therefore, this fivestep with ssprk5 method is also favoured by many users. But since the coefficients of ssprk5 are too long to present here, the users are recommended to see the Appendix B in Spiteri et al. 2002 or Section 2.1 in Gottlieb 2005 for details (but actually, the coefficients in these two papers are not exactly the same...) .