Time Integration

Time integration advances the solution of the semi-discrete system obtained after spatial discretization of the governing equations. In finite volume form, the governing equations can be written as:

$$\frac{dq}{dt} = f(q)$$

where $q$ is the vector of conserved variables and $f(q)$ represents the spetial residual including non-linear terms such as convective fluxes, viscous fluxes, and source terms.

Available Time Integration Schemes

In CompFlowLab, the time integration scheme can be specified by time_scheme in input file. Following are some of the available time integration schemes.

FDF: Forward Difference Euler (Explicit)

A first-order explicit method:

$$q^{n} = q^{n-1} + \Delta t \, f(q^{n-1})$$

• First-order accurate
• Simple and inexpensive
• Strongly limited by CFL condition

SSPRK2: Strong Stability Preserving Runge–Kutta 2 (Explicit)

A second-order explicit method:

$$q^{(1)} = q^{n-1} + \Delta t \, f(q^{n-1})$$ $$q^{n} = \frac{1}{2} q^{n-1} + \frac{1}{2} \left( q^{(1)} + \Delta t \, f(q^{(1)}) \right)$$

• Second-order accurate
• Improved stability over Forward Euler
• Suitable for moderate accuracy requirements

SSPRK3: Strong Stability Preserving Runge–Kutta 3 (Explicit)

A second-order explicit method:

$$q^{(1)} = q^{n-1} + \Delta t \, f(q^{n-1})$$ $$q^{(2)} = \frac{3}{4} q^{n-1} + \frac{1}{4} \left( q^{(1)} + \Delta t \, f(q^{(1)}) \right)$$ $$q^{n} = \frac{1}{3} q^{n-1} + \frac{2}{3} \left( q^{(2)} + \Delta t \, f(q^{(2)}) \right)$$

• Third-order accurate
• Widely used in high-resolution CFD
• Good balance of accuracy and stability

BDF: Backward Difference Euler (Implicit)

A first-order implicit method:

$$q^{n} = q^{n-1} + \Delta t \, f(q^{n})$$

• First-order accurate
• Unconditionally stable for many problems
• Requires solving a nonlinear system each time step

The BDF method uses finite differences to compute the Jacobian in reacting flow cases, which might make it very stiff and hard to converge if the time step is too large.