Stability time step restrictions

The numerical approach is named the pseudo-spectral method because the integration with respect to time is carried out in the physical space, and the Fourier transform related quantities which are solved in spectral space are used only for the calculation of the spatial derivatives. The main advantage of the PsM compared to the FDM is the low cost of the computations, as for the same accuracy the PsM required a much smaller number of space grid points. Moreover, when using the PsM, the DFT of the variable/ is computed at every discrete time step. The major disadvantage of the method is the time step restrictions that must be imposed to maintain stability due to the explicit time integration scheme employed. Another disadvantage of the Fourier PsM is related to the periodic boundary conditions required. However, as mentioned earlier, non-periodic boundary conditions can be incorporated if Chebyshev polynomials are used as basis functions. 

Above, we have studied in detail the stability properties of simple single-step methods of the class (4.150). More generally, explicit methods always have time restrictions of the form (4.169), and the more a method makes use of past data to predict the fiiture, the more strict the time step restrictions become. We consider here the stabiUty properties of implicit multi-step BDF methods... 

Computational issues that are pertinent in MD simulations are time complexity of the force calculations and the accuracy of the particle trajectories including other necessary quantitative measures. These two issues overwhelm computational scientists in several ways. MD simulations are done for long time periods and since numerical integration techniques involve discretization errors and stability restrictions which when not put in check, may corrupt the numerical solutions in such a way that they do not have any meaning and therefore, no useful inferences can be drawn from them. Different strategies such as globally stable numerical integrators and multiple time steps implementations have been used in this respect (see [27, 31]). 

In contrast to the method of characteristics, which gives faithful simulation of transient flows but which is very restrictive in time step sizes, the stability of the implicit methods permit large time steps and drastic reduc-... 

Implicit methods are a bit more complicated to implement, but they are highly stable compared to explicit methods. For a linear system of equations, such as the present problem, there is no stability restriction at all. That is, the method will produce stable solutions for any value of the time step, including dt - oo. For nonlinear problems, or for higher-order time differencing, there is a stability limit. However, the implicit methods are always much more stable than their explicit counterparts. 

In addition to the restrictions that stability places on the step size hn, we also need to be concerned with how accuracy affects the choice of step size. Assume that the local accuracy is to be controlled to within a certain tolerance e and that accuracy can be estimated by the local truncation error. The time step must be chosen to keep a norm of the local truncation error below the tolerance, that is ... 

The explicit method is easy to use, but it suffers from an undesirable feature that Severely restricts its utility the explicit method is not unconditionally stable, and the largest permissible value of the lime step At is limited by the stability criterion. If the time step At is not sufficiently small, the solutions obtained by the explicit method may oscillate wildly and diverge from the actual solution. To avoid such divergent oscillations in nodal temperatures, the value of Af must be maintained below a certain upper limit established by the stability criterion. It can be shown mathematically or by a physical argument ba.sed pfl thc second law of thermodynamics tliat the stability criterion is satisfied if the coefficients of alt in the Tjj, expressions fcalled the primary... 

Different equations for different nodes may result in different restrictions on ihe ize of the time step Af, and the criterion that is most restrictive should be used in the solution of the problem. A practical approach is to identify the equation with the smallest primary coefficient since it is the most restrictive and to determine the allowable values of At by applying the stability criterion to that equation only. A At value obtained this way also satisfies the stability criterion for all other equations in the system. 

Next we need to determine the upper limit of the time step Af from the sta-bilily criterion since we are using the explicit method, This requires the iden-tiflcalfbn of the smallest primary coefficient in the system. We know that the bouiTdary nSdes are more restrictive than the interior nodes, and thus we examine the formulations of the boundary nodes 0 and 5 only. The smallest and thus the most restrictive primary coefficient in this case is the coefficient of Tq in the formulation of node 0 sitree 1 - 3.74t < J - 2.7r, and thus the stability criterion for this problem can be expressed as... 

For physically realistic and bounded results, it is necessary to ensure that all the coefficients of the discretization equation are positive. This requirement imposes restrictions on the time step that can be used with different values of 0. It can be seen that a fully implicit method with 0 equal to unity is unconditionally stable. Detailed stability analysis is rather complex when both convection and diffusion are present. In general, simplified criteria may be used when an explicit method is used in practical simulations ... 

Although the grid chosen was coarse the agreement with the explicit solution from (2.171) is satisfactory. However the first 12 time steps record only a small part of the cooling process. This is due to the restriction of the time step At in the stability condition. This can only be overcome by transferring to an implicit difference method. 

After rearranging Eq. (4.59) for 6"+1, the same criterion can again be recovered by setting the coefficient of 0" positive. We learn from this result that the presence of enthalpy flow further restricts the size of the time step for a given spatial discretization. A constant source term, say u " jk, however, has no effect on the stability of the numerical... 

We aim at the development of fully robust, stable methods and therefore we restrict our attention to implicit methods with Particularly, we shall consider two cases with 6 = Yi and 0 = 1, which correspond to the Crank - Nicolson (CN) and backward Euler (BE) method, respectively. More details can be found e.g. in Quarteroni Valli (1994). Finally, we chose the BE method for its higher stability (the CN scheme can show some local oscillations for large time steps). 

Many direct numerical simulations (DNS) for combustion flows were developed considering constant density [13], Constant density increases the stability of a numerical code considerably. But, these formulations are limited by the absence of flame-induced flow modifications due to heat release. On the other hand, if the fully compressible equations are employed to solve low Mach number flows, the high-frequency acoustic waves create severe restrictions on the time-stepping increment. 

A physical interpretation of the above equation is that the maximum step in time is up to a numerieal faetor the diffusion time across a spatial cell of width Ax. The restriction on the allowable time step for stability is very severe in most practical problems as times of interest are typically much larger than the maximum allowable time step. Note that as the spatial resolution increases, the requirement on the time steps beeome very small. As previously discussed with regard to single variable differential equations, the FD method is of little practical use in solving partial differential equations. 

The restriction on the step size (2.304) due to the stability condition for the explicit difference method can be avoided by using an implicit method. This means that (2.298) is discretised at time tk+1 and the backward difference quotient is used to replace the time derivative. With... 

Such stability depends to a significant degree on the interrelation of discretization steps for the forecast object in space and time. There are certain restrictions of such interrelations. If this restriction is exceeded, numerical... 

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