Multi-Staging approach (MUSTA)

When considering a numerical scheme, there are two types of accuracy that need to be considered.

Firstly, the order of accuracy, which determines the rate at which the numerical solution tends to the exact solution with increasing resolution. Secondly, the overall accuracy, which is the actual difference between the exact and computed solution. The point to note is that it is possible to increase the overall accuracy of a method without increasing itsorder of accuracy.

In order to improve on the accuracy of SLIC, we would usually consider Riemann solvers, which use information about the wave types making up the solution to calculate the flux. These tend to be computationally intensive, often requiring iterative or matrix methods to determine the state at the cell

x/t= 0

i−¹2 i+¹₂ i+³₂

Figure 4.3: How to multi-stage a simple numerical scheme (e.g. FORCE). The Riemann problem is solved for several time steps using the simple scheme on the two-cell domain, with transmissive boundary conditions as shown by the empty arrows. The solid lines show the wave structure of the solution. As the number of stages increases, the waves move away from the centre, so that the flux at the centre tends to the correct limit, which is represented by the filled arrow at the top. This figure corresponds to calculating the FORCE³ scheme, if FORCE is used as the underlying simple scheme.

face. In order to increase the order of accuracy, we could use a higher-order reconstruction than the linear one that SLIC uses. Again, this would require more computational expense.

However, Toro [110] has suggested a simple way of improving the accuracy of any numerical scheme, which can produce improved accuracy for less computational effort than would be required if the accuracy were to be attained by increasing resolution. This is called MUSTA, which stands for MUlti-STAging and can be understood as follows:

The FORCE flux provides a simple way of estimating the flux between two cells, given a Riemann problem. However, if we were to evolve a Riemann problem for a few time steps, the exact flux between the two cells would remain constant due to the similarity of the solution in x/t, and the values of the solution immediately to either side of the initial discontinuity would become closer. The FORCE flux can provide a better estimate of the correct flux given two close states than two separated states, and so its approximation of the central flux would be improved. Therefore, the MUSTA approach is to evolve the Riemann problem on a two cell grid, with transmissive boundary conditions, for a few time steps, or stages, and to take the central flux at the last stage as the estimate for the final flux. This can be represented as in Figure 4.3. In terms of an evolution algorithm, multi-staged FORCE can be written as the iterative scheme

f_i+^FORCE1 ^k

2 =f_i+^FORCE1 2

¡u^k_L⁻¹,u^k_R⁻¹¢

, (4.23)

u^k_L=u^k_L⁻¹− ∆t

∆x

³

f_i+^FORCE1 ^k

2 −f(u^k_L⁻¹)´

, (4.24)

u^k_R=u^k_R⁻¹− ∆t

∆x

³

f(u^k_R⁻¹)−f_i+^{FORCE1 k}

2

´, (4.25)

whereu⁰_L=uL andu⁰_R=uR give the initial conditions.

The best improvements in accuracy for least computational expense are likely to be found when multi-staging is applied to as simple a scheme as possible. Therefore, in order to improve the accuracy of SLIC, we do not multi-stage SLIC itself, but rather multi-stage the FORCE scheme on which SLIC is based. We write the multi-staged FORCE scheme as FORCE^k, wherek is the number of stages, so that FORCE¹≡FORCE. This notation is as in [110] but differs from that in [112].

Although the preceding explanation seems plausible enough, we can examine the theoretical effect of MUSTA for the linear advection equation. As mentioned in§4.2.1, the FORCE scheme can be expressed as

f_i+^FORCE1

2 =ωf_i+^LW1

2 + (1−ω)f_i+^LF1

2 withω= ¹₂. (4.26)

The local CFL numbercL is derived from thelocal wave speeda_i+¹

2 andglobal time step ∆tas cL= ∆t

∆xa_i+¹₂. (4.27)

For linear advection, all numerical schemes with a centred stencil of width at most three can be written in the form (4.26), and so we can plot the dependence ofωoncLfor various schemes, as in Figure 4.4. For the linear advection problem, other schemes can be plotted on the same diagram. The exact Riemann solver (Godunov scheme) falls on the curved line shown. This splits the diagram into two parts. Above the curve lie schemes that are oscillatory, and below it lie schemes that are non-oscillatory, but are more diffusive than the ideal. The exact Riemann solver is the least diffusive non-oscillatory first-order method for the linear advection equation. The FORCE scheme can therefore be seen to be the least diffusive non-oscillatory first-order method not to depend on the local wave speed information given bycL.

Although the MUSTA approach does not depend explicitly on cL, it does depend on it implicitly, through its dependence on the local data for the predictor step (4.23). This dependence can be calculated in a straight-forward, but algebraically intensive manner, which we have done using Maple^TM, and we plot the results for FORCEⁿ, n= 1, . . . , 5 in Figure 4.5. It is immediately clear that MUSTA may not be a suitable evolution scheme at high CFL numbers, even for such a simple equation as linear advection, as all of the schemes now protrude into the oscillatory region. Using very high numbers of stages shows that FORCE^k tends to the Lax-Wendroff scheme ask→ ∞(except forcCFL= 1 whereω=¹₂ for allk, so that the convergence is not uniform).

However, some improvement can be made to this. If we replace ∆t by α∆t all the way through, i.e. in equations (4.24),(4.25) and (4.12), keeping α the same throughout a multi-staging calculation, then using the values ofαin Table 4.1, we find the situation in Figure 4.6. Here we see that there is no protrusion into the oscillatory region, and that we have reduced the diffusivity of the FORCE scheme.

0 0.2 0.4 0.6 0.8 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c L

ω

Lax−Friedrichs FORCE

Lax−Wendroff

Godunov

Figure 4.4: Four basic schemes for evolving the linear advection equation, displayed in terms of the local CFL number cL, which affects the weighting used to construct the method from the Lax-Friedrichs and Lax-Wendroff schemes.

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

c

ω

Lax−Wendroff

FORCE²

FORCE¹

Lax−Friedrichs Godunov

Figure 4.5: Same as Figure 4.4, with additional multi-staged versions of FORCE^k where k = 1, . . . ,5 andk increases up the page (labels fork= 3,4,5 are omitted for reasons of clarity).

However, increasing the number of stages beyond two is of little use. The values forα were calculated using Maple^TM, the condition being that the curves should lie below the Godunov curve for all values of

0 0.2 0.4 0.6 0.8 1 0.5

0.55 0.6 0.65

c

ω

Godunov FORCE²

FORCE¹

Figure 4.6: Same as Figure 4.5, except that ∆t in (4.24) and (4.25) has been scaled by a factor α to maintain the monotonicity of the scheme overall. Note the expanded ω scale as compared to the other two figures.

stages α

1 not applicable

2 ¹₂

3 1−^√1₂ 4 1− ^√31₂ 5 1− ^√41₂

Table 4.1: Values of αthat make a multi-staged FORCE scheme stable for the linear-advection problem when replacing ∆t byα∆t.

cCFL. We have not been able to prove the apparent relation thatα= 1−2^1−k in general.