2.5 Properties of the Interface Schemes
2.5.3 Time-Marching Stability
In this section we consider the fully discrete version of equation (2.2.1), having chosen a time-marching scheme for the semidiscrete equation (2.2.25). The stability analysis for explicit Runge-Kutta methods in the uniform grid case is presented in Appendix A.2, but here it is generalized to nonuniform discretizations.
First, the semidiscrete equation is written in terms of a generalized difference matrix D=P−1Q,
du
dx =−c 1
∆xDu=Au, (2.5.7)
the right-hand side of which is in the form of equation (2.2.6). Following Appendix A.2, the time-marching scheme has the diagonalization of S, where
S=XΛSX−1, (2.5.8)
where ΛS = diag λS
= diag σ(λA)
. From equation (2.5.7), the eigenvalues of A are related directly to the eigenvalues of the difference scheme by
λA=− c
∆xλD. (2.5.9)
Thus stability is directly dependent on the eigenvalues of the difference matrixD,λD. This gives the discrete stability criteria following equation (A2.5),
|σ(−CλD)|<1, (2.5.10) where the time increment has been written in terms of the CFL number
C = c∆t
∆x. (2.5.11)
This result gives the discrete stability bound for a given difference matrixD. Note that any change to the grid will change the form of D and potentially its limiting eigenvalue, so care is required when defining the stability bound on the CFL number. Also, for a grid with periodic boundaries and a centered finite-difference scheme, the eigenvalues λD are purely imaginary. The following sections investigate the discrete stability bounds for the fourth-order explicit scheme and the corresponding interface solution from Section 2.3.2, with respect to the third-order RK32 Runge-Kutta time-marching scheme from Butcher (2003) used for the results in Section 2.6.
2.5.3.1 Stability for Uniform Grids
To put the stability limits for grids with local refinement in a proper context, we first examine the uniform grid examples shown in Table 2.11. Four grids of 100 points each were considered, this being enough points for the periodic boundary case to closely approach the theoretical CFL limit for an explicit fourth-order finite-difference scheme with a third-order
Runge-Kutta scheme (determined using the spectral analysis presented in Appendix A.2).
The results show that finite boundaries do not affect the CFL stability limit, for values of the SAT parameterτ small enough. For the τ = 2 case, the spectrum of which is shown scaled by the limiting CFL in Figure 2.4, all eigenvalues have negative real parts and stability is limited by the eigenvalues with the largest imaginary part. For largerτ, stability becomes constrained by the eigenvalue with the largest negative real part, which significantly reduces the CFL limit compared to the periodic case.
Grid CFL limit
Theoretical, periodic boundaries 1.2622 Finite matrix, periodic boundaries 1.2627
SAT boundaries,τ = 2 1.2630
SAT boundaries,τ = 4 0.3765
SAT boundaries,τ = 6 0.2193
Table 2.11: Stability bounds for a uniform grid with the fourth-order explicit finite-difference scheme. The theoretical value for a periodic grid is obtained by spectral analysis, and the remaining values from the minimum eigenvalue of a 100-point domain.
!0.15 !0.10 !0.05 0.00 0.05
!2
!1 0 1 2
Real part
Imaginarypart
Figure 2.4: Eigenvalue spectrum for a 100-point uniform grid using the fourth-order explicit scheme and SAT boundary conditions withτ = 2. Values are scaled by the limiting CFL to lie within the blue line indicating the stability boundary for the third-order Runge-Kutta scheme.
2.5.3.2 Stability for Nonuniform Grids
Two different grids, each of 100 points, are considered to explore the effect of local refinement on time-marching stability, with results shown in Table 2.12. The first grid has a central refinement region, the domain being split into three blocks of 30, 40 and 30 points each.
The central block of 40 points has a discretization of ∆x, with the other two blocks having 2∆x, for a refinement scheme 2:1:2. This arrangement permits this grid to have periodic boundary conditions, in which case the eigenvalues are (as expected) purely imaginary and the CFL stability limit is noticeably reduced compared to the uniform grid case. With finite boundaries implemented by the SAT method withτ = 2, shown scaled by the limiting CFL in Figure 2.5, the spectrum has eigenvalues with nonpositive real parts (some have zero real parts), but there is no change to the limiting eigenvalue on the imaginary axis and the stability limit is the same as the periodic case. Usingτ = 6 brings a small reduction to the CFL limit, but the limiting eigenvalue is now on the real axis and is, in fact, the same as in the uniform grid case. See that 0.4387/2≈0.2193, the factor of two resulting from the boundary region having discretization 2∆xin this case compared to ∆xin the uniform grid case.
The second grid considered has two blocks of 50 points each, the first of discretization ∆x and the second 2∆x, for a refinement scheme 1:2. This grid cannot have periodic boundary conditions as the boundaries have inconsistent discretizations, so the SAT boundary imple- mentation is used instead. With τ = 2, shown scaled by the limiting CFL in Figure 2.6, there is no stability reduction compared to the uniform case due to the interface, and the eigenvalues have all negative real parts. The interface again has no effect compared to the uniform case withτ = 6.
Grid CFL limit
Three-block refinement, periodic boundaries 0.4851 Three-block refinement, SAT boundaries, τ = 2 0.4851 Three-block refinement, SAT boundaries, τ = 6 0.4387 Two-block refinement, SAT boundaries,τ = 2 1.2652 Two-block refinement, SAT boundaries,τ = 6 0.2193
Table 2.12: Stability bounds for various nonuniform grids with the fourth-order explicit finite-difference scheme. Each limit is calculated from the minimum eigenvalue of a 100- point domain and based on the smallest discretization. See text for a detailed description of each grid configuration.
!0.15 !0.10 !0.05 0.00 0.05
!2
!1 0 1 2
Real part
Imaginarypart
Figure 2.5: Eigenvalue spectrum for the three-block (2:1:2) 100-point grid, using the fourth- order explicit scheme and SAT boundary conditions with τ = 2. Values are scaled by the limiting CFL to lie within the stability boundary of the third-order Runge-Kutta scheme (blue line). Here, Cmax= 0.4851.
!0.15 !0.10 !0.05 0.00 0.05
!2
!1 0 1 2
Real part
Imaginarypart
Figure 2.6: Eigenvalue spectrum for the two-block (1:2) 100-point grid, using the fourth- order explicit scheme and SAT boundary conditions with τ = 2. Values are scaled by the limiting CFL to lie within the stability boundary of the third-order Runge-Kutta scheme (blue line). Here, Cmax= 1.2652.
The conclusion to be drawn from this investigation is that stability of the discrete prob- lem with local refinement is strongly dependent on the form of the grid. The presence of grid interfaces can play a secondary role compared to the implementation of the bound- ary conditions, but reduction of the CFL limit due to local refinement can be significant for certain schemes. These results are also dependent on the time-marching scheme used;
for methods other than Runge-Kutta, the presence of refinement may have a more signif- icant effect or none at all. For the validation problems considered in Section 2.6 and the Richtmyer-Meshkov problem of Chapter 4, a general rule of thumb was that C . 0.2 for stability on most grids. Instability due to the CFL number usually manifested itself at the boundaries first.