Iteration Parameters

The choice of the iteration parameters are essential for the accelerated PT method as the method is highly sensitive to it.

Choice of iteration parameters

Generally, the values of the iteration parameters are associated with the maximum eigenvalue of the stiffness matrix.

The optimal iteration parameters for a variety of basic physical processes can be determined analytically but for most of the practical problems the eigenvalue problem needs to be solved.

For the eigenvalue problem see more about its numerical aspects here.

Factor 1: Choice of the B.C.

Factor 2: Numerical stability restriction

• for explicit time integration, the size of the timesteps is upper-bounded

Case studies

Physical processes

1). Diffusion

\[\rho \frac{\partial H}{\partial t} = -\nabla_k q_k\]

\[q_i = -D \nabla_k H, i = 1... n_d\]

Or by plugging the second equation into the first one we obtained a single equation for describing the diffusive process.

\[\rho \frac{\partial H}{\partial t} = \nabla_k (D \nabla_k H)\]

• H ↔ some physical quantity

• D ↔ diffusion coefficient

• ρ ↔ proportionality coefficient

• t ↔ physical time

The stationary diffuion process is given by the above equation when $\frac{\partial H}{\partial t}\rightarrow 0$

\[0 = \nabla_k (D \nabla_k H)\]

Applying PT method

For the accelerated PT method we do the following:

• STEP 1: add the inertia term $\theta_r \frac{\partial q_i}{\partial \tau}$ to the LHS of the first equation

• STEP 2: plug the obtained equation from step 1 into the equation 2 to obtain the damped wave equation.

  • PDE type switch from parabolic to hyperbolic

  • describes also wave propagation

\[\tilde{\rho} \theta_r \frac{\partial^2 H}{\partial t^2} + \tilde{\rho} \frac{\partial H}{\partial \tau} = \nabla_k (D \nabla_k H)\]

• STEP 3: find the optimal Reynolds number

  • \[Re = \frac{\tilde{\rho}V_p L}{D}\]

    , where $V_p = \sqrt{\frac{D}{\tilde{\rho} \theta_r}}$ is the finite speed of the information signal of the wave propagation.

  • This can be done by the dispersion analysis, the optimal value of $Re$ in this case is $Re_\text{opt} = 2 \pi$

• STEP 4: obtain the optimal parameters of $\tilde{\rho}, \theta_r$ using the optimal Reynolds number

  • Generally: "Low Re ⇒ flows tend to be laminar" and "High Re ⇒ flows tend to be turbulent"

\[\tilde{\rho} = Re \frac{D}{\tilde{V}L}\]

\[\theta_r = \frac{D}{\tilde{\rho} \tilde{V}^2} = \frac{L}{Re \tilde{V}}\]

Here we need to solve for $V_p = \tilde{V}$, where $\tilde{V} := \frac{\tilde{C}\Delta x}{\Delta \tau}$ is the numerical velocity we just introduced. Note that $\tilde{C} \approx 0.95 C$ is used here, which is an emperically determined parameter deduced from numerical experiments.

In case $D := D(x_k)$ is not constant, we need to determine its values locally to each grid point. For particularly discontinuous distribution of $D$, taking a local maximum of $D$ between two neighbouring grid cells for determining the iteration parameters shall be sufficient. Räss et al

• STEP 5: perform explicit time stepping

Restriction for the size of the pseudo timestep damped wave equation: $\Delta \tau \leq \frac{C}{V_p} \Delta x$, where $\Delta x = \frac{L}{n_x}$, value of the non-dimensional number $C$ is determined for the linearised problem (von Neumann stability analysis)

Results

The number of iterations required for the method to converge is linearly dependent on the numrical grid resolution $n_x$