Dual time stepping implementation

Dual time stepping interest. Aeroelasticity and fliid-structure coupling computations are usually performed using a very small global time step value. This is especially true when studying low frequency phenomena. Therefore a very large number of iterations is required, and this leads to very expensive computations. In the viscous case, large and very refined meshes are used, and the time step requirements for numerical stability are even more critical. Moreover, moving meshes computations are required, which increases CPU costs.

This is the reason why the use of dual time stepping for Navier-Stokes aeroe – lastic computations becomes very interesting. The physical time step used to describe the unsteady phenomenon is no longer constrained by stability time step values in the smallest cells. At each physical time step, a modified steady problem is solved in a dual pseudo time. Usual convergence acceleration tech­niques such as local time stepping or multi-grid scheme may be used. Dual local time steps are bounded by specific stability requirements.

As far as moving meshes computations are concerned, the dual time step technique helps to reduce the number of remeshing computations. Dual time iterations are performed at a fixed physical time step, that is to say in a fixed mesh. This is much more important in the case of coupled fhid structure com­putations, where the position and velocity of the grid is not prescribed, but derives from the resolution of the coupled equations.

1.1.1 Dual time stepping scheme for moving meshes. Let us consider

du

equation (E) : — + /(u) = 0, where и is a numerical function of time. Writ­ing 2nd order Taylor’s expansions for u(tn) and u(tn-1) at time tn+1 allows us to write equation (E) at time tn+1 as follows :

du* u(t„_i) — 4u(t„) + 3u* ~dt* H 2At

+ f (u*) + o(At)

u(tn+1) is then obtained as the solution at steady state of a differential equation for the variable u*, function of the pseudo-time t*:

Pseudo-time t* is called dual time. The resolution of the unsteady problem is now performed within a system of two time loops. The external one is the physical time loop. The inner one is the dual time loop. The dual time loop is carried out using local time stepping, because it solves a steady state problem. The usual four step Runge-Kutta scheme is used to perform the dual loop res-

Q*(k) = Q

with Q*(0)

At* ■

01 k у n+1

Vn-1Qn-i – 4VnQn + 3Vn+1Q*(k-1) ]

= Qn, Q*(n+1)

= Q*(4),

olution. Within this loop, the grid is fixed at physical time tn+1. Applying this approach to the conservative fliid equations in moving grid, we obtain:

RVq-1) = ^ F(q-1) NSi – VTV0) and F(q-1) = F(q-1) + Fd(q-1) + D(0)

i=1

In these formulae, At and At* denote the physical and dual time steps, respectively. At steady state of the dual loop, the conservative variables vector Q is obtained at time t(n+1). The stability condition for the dual time scheme depends on the value of the physical time step as follows:

4

At* < CFL x – At 3

This condition is added to the those given by the properties of the Jacobian matrices of the convective and viscous flixes.