Temporal Discretization for Transient Problems

Numerical simulations of transient flow problems require robust temporal discretization techniques. In CMPS, implicit and explicit methods are available for solving these problems. Each method has its specific applications, advantages, and limitations.

Implicit vs. Explicit Methods

Explicit methods are primarily used for problems involving acoustic wave propagation. However, they come with several limitations, such as stringent stability constraints that often require very small time steps.

Implicit methods, on the other hand, are more versatile and stable, particularly for stiff problems or scenarios involving large time scales. These methods allow larger time steps and are well-suited for problems with strong nonlinearities or multi-scale phenomena, such as chemical reactions and thermal processes.

While implicit methods excel in stability, their accuracy may degrade for large time steps. First-order time discretization with small time steps can still outperform explicit methods in many scenarios. Choosing the correct time step size is critical and often problem-dependent.

First- and Second-Order Temporal Discretization

Temporal discretization can be performed using first-order or second-order schemes, each offering different levels of accuracy and computational cost.

First-Order Temporal Discretization

First-order schemes are computationally efficient and simpler to implement but less accurate. The error decreases linearly with the time step size \( \Delta t \). In CMPS, the first-order finite difference method is expressed as:

\[ \frac{\partial Q}{\partial t} = \frac{Q^{n+1} - Q^n}{\Delta t}, \]

where \( n \) represents the current time step. Although these schemes are less accurate, they can be advantageous for problems where computational efficiency is a priority.

Second-Order Temporal Discretization

Second-order schemes offer improved accuracy by reducing errors quadratically with the time step size \( \Delta t^2 \). CMPS employs the second-order backward difference method by default, given as:

\[ \frac{\partial Q}{\partial t} = \frac{3Q^{n+1} - 4Q^n + Q^{n-1}}{2\Delta t}. \]

These schemes are particularly effective for capturing system dynamics with larger time steps, but they require more computational resources compared to first-order methods.

Choosing Between First- and Second-Order Methods

The choice of temporal discretization depends on the balance between accuracy and computational cost. First-order methods are suitable for extensive simulations where efficiency is paramount, whereas second-order methods are preferred for high-accuracy applications.

Dual Time Stepping

Dual time stepping is an advanced technique used to enhance the stability and efficiency of transient simulations. It introduces a pseudo time step \( \Delta \tau \), distinct from the physical time step \( \Delta t \).

The pseudo time step is used to iteratively solve the discretized equations at each physical time step, allowing for rapid convergence. This approach is particularly useful for low-speed and compressible flows, where preconditioning plays a critical role.

In CMPS, dual time stepping restores time accuracy that may be lost due to preconditioning. The pseudo time integration uses a first-order method, while the physical time integration can employ first- or second-order schemes. The governing equation for physical time integration in CMPS is:

\[ \frac{\partial W}{\partial t} = \frac{3W^{n+1} - 4W^n + W^{n-1}}{2\Delta t}. \]

Here, \( W \) represents conservative variables, as preconditioning is applied only to primitive variables \( Q \).

Deferred Correction Method

In CMPS, a deferred correction (DC) method is used to improve temporal accuracy with implicit methods. This approach combines lower-order solutions with error corrections to achieve higher-order accuracy at reduced computational cost.

The general process involves solving the equation:

\[ \frac{dy}{dt} = f(t, y), \quad y(t_0) = y_0, \]

using an implicit approximation at each time step. The initial approximation at \( t_{n+1} \) is given by:

\[ y_{n+1}^{(0)} = y_n + \Delta t \cdot f(t_{n+1}, y_{n+1}^{(0)}). \]

Refinements are made using iterative methods such as Newton-Raphson, and error estimates are applied to correct the solution. The deferred correction process ensures higher-order accuracy by interpolating corrections across the solution space.

This method is particularly effective in CMPS, where first-order solutions are refined to achieve second-order accuracy without the full computational cost of a second-order scheme. Adaptive sub-stepping ensures stability while maintaining efficiency.