Segregated Runge–Kutta methods
The Segregated Runge–Kutta method is a family of IMplicit–EXplicit Runge–Kutta methods that were developed to approximate the solution of differential algebraic equations of index 2.
The SRK method were motivated as a numerical method for the time integration of the incompressible Navier–Stokes equations with two salient properties. First, velocity and pressure computations are segregated. Second, the method keeps the same order of accuracy for both velocities and pressures. However, the SRK method can also be applied to any other DAE of index 2.
The Segregated Runge–Kutta method
Consider an index 2 DAE defined as follows:In the previous equations is known as the differential variable, while is known as the algebraic variable. The time derivative of the differential variable,, depends on itself,, on the algebraic variable,, and on the time,. The second equation can be seen as a constraint on differential variable,.
Let us take the time derivative of the second equation. Assuming that the function is linear and does not depend on time, and that the function is linear with respect to, we have that
A Runge–Kutta time integration scheme is defined as a multistage integration in which each stage is computed as a combination of the unknowns evaluated in other stages. Depending on the definition of the parameters, this combination can lead to an implicit scheme or an explicit scheme. Implicit and explicit schemes can be combined, leading to IMEX schemes.
Suppose that the function can be split into two operators and such that
The SRK method is based on the use of IMEX Runge–Kutta schemes and can be defined by the following scheme: