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General Time Integration Schemes

2-Parameter Time Integration

    With the spatial terms of the quasi 1-D Euler equations dealt with in the previous section, the temporal term must be dealt with.  A two parameter family of time integration schemes may be defined by eqn. (13), where the delta symbol represents a forward difference stencil -- the difference between a variable at the n+1 time level and the n time level.  Likewise, the nabla operator is a backward differencing stencil -- the difference between the n time time level and the n-1 time level.
   

(13)

This general time differencing stencil can be equated to the residual defined in the previous section in eqn. (12).

(14)

The operators can be manipulated for simplicity to reflect eqn. (15).

(15)

Since the quantities at the n time level are known, the unknowns at the n+1 time level can be isolated as shown by eqn. (16).  Solving the time integration scheme in terms of the forward difference stencil rather than the n+1 time level directly is preferred in CFD applications because the difference stencil can be directly related to the residual and offers a simple, straight-forward formulation.  The forward difference of the residual is also written out in eqn. (16).

(16)

Since the residual at the n+1 time level is not known, linearlization is applied through the use of a Taylor Series expansion.  Equation (17) results in the residual at the n level plus the Jacobian matrix of the residual at the n level time the forward difference stencil of the conservative variable vector.  Equation (17) highlights two important ideas.  First, the conservative variable vector is denoted by a vector arrow as a explicit reminder that it is not a scalar.  Even though the vector notation was dropped at the beginning the avoid clutter, it is noted here.  Second, a summation is applied in the linearization term since the n+1 level residual is, in general, a function of the variables at every point in the discrete grid.  For the MUSCL spatial stenciling used by Gryphon in particular, the residual at each grid point happens to be a function of the conservative variables at that grid point and the two grid points immediately to each side.  Thus, the summation here will actually be only for j-2, j-1, j, j+1, j+2 for each grid cell and form a pentadiagonal matrix.  

(17)

The residual linearization can be substituted into eqn. (16) to form eqn. (18).

(18)

The cancellation of the n level residual terms can be accomplished in the last term, and the residual Jacobian term moved to the left hand side to finalize the general time integration form.  All things in principle are thus known in eqn. (19) except the forward conservative variable stencils.

(19)

If the theta parameter is zero, the method is a fully explicit method and the solution at each point is fully decoupled from all other points allowing for easy solution; otherwise, it is an implicit method and a matrix solution is required.Implicit methods require more effort to solve each iteration, but larger timesteps can be taken since there is no timestep restriction as there is with explicit methods.  Any method with a zeta parameter that is not zero is a three point time stencil method, requiring the solution to be stored at the n+1, n, and n-1 levels.  This requires more storage and somewhat slower execution time, but can offer greater temporal accuracy.
    A variety of methods can be defined using these two parameters.  Several common choices that are employed by Gryphon are given in Table 1 below.

Table 1. Two Parameter Time Integration Schemes as Programmed in Gryphon
	theta	zeta	local time accuracy
Euler Implicit	1	0	dt squared
Euler Explicit	0	0	dt squared
Leap - frog	0	- 1/2	dt cubed
Trapezoidal	1/2	0	dt cubed
3-point backwards	1	1/2	dt cubed

Runge-Kutta Time Integration

    Another common choice of time integration techniques is known as Jameson-style M-stage Runge Kutta.  This technique is based on the common method of solving ordinary differential equations with some slight adaptation for the purpose here.  Equation (20) outlines the scheme.  The modification comes in the fact that the solution at each stage is always with respect to the original n-level solution.  This streamlines the integration process as only the original n-level and the most current solution needs to be saved in memory.  At each stage, the residual using the most current solution available is calculated as normal, and then a new stage solution is calculated by eqn. (20).  At the end of integration, the n+1 level solution is simply the last stage calculation.  Any value of M may be used, but M = 4 is the most common and is the type of Runge-Kutta implemented automatically in Gryphon.  M-stage Runge Kutta is a fully explicit method.

FOR k = 1 to M

(20)

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