Ansys Forte employs an advanced operator-splitting method to solve the species-conservation and energy-conservation equations for time-accurate transient simulations. This method splits the transport equation into two sub-equations and solves the sub-equations with overlapping time-steps; the first step handles the kinetics and the second step handles the transport (for example, diffusion or convection). The chemistry solver described in this section handles the first step. For this step, the calculation is performed on a cell-by-cell basis, allowing coupling of all of the species and energy together, since the transport terms are handled only in the second step. The transport of the species (that is, the second step) is then handled using the methods described in SIMPLE Method . With the source term being handled by an advanced chemistry solver, the CFD step is well behaved and can proceed using reasonable time-step values that are limited only by the desired transport accuracy.

To illustrate the method using equations, we first define the full species equation that we want to solve in each cell in the computational domain as:

(5–8)

where is the net production rate of species k due to all chemical reactions that occur in that cell, V _cell is the volume of the cell, ρ is the mass density of the fluid, Y _k is the species mass fraction, C _j represents the convective mass flow of the species into a cell from neighboring cell j , and D _j is the diffusive mass flow of the species into a cell from the j th neighboring cell. Note that this equation is simplified just to illustrate the operator-splitting methodology.

We can approximate the time derivative on the left-hand-side of Equation 5–8 , for a discrete time step, t , so that the equation becomes:

(5–9)

where the superscript " n " represents evaluation of the terms at the new time and " o " represents evaluation at the old time. With the convective terms and the production terms all evaluated at the new time, the equation is fully implicit and fully coupled within the cell. Equation 5–9 can be rearranged to solve for the new value of the species mass fraction as:

(5–10)

Now, since it is extremely difficult to solve this equation exactly, we choose to split the equation by letting:

(5–11)

(5–12)

In this way, we lose some implicitness in the chemistry production term, but we gain the fact that Equation 5–11 is now solved for each cell independently with no influence of the neighboring cells. This allows efficient solution of all of the species simultaneously instead of equation by equation. With operator splitting, the first step balances production and destruction of chemical kinetics against the time-rate-of-change of the species in a cell, neglecting transport into or out of the cell. In this step the solution algorithm couples all species calculations together within each cell. In this calculation, there is no transport into or out of the cell, such that each cell can be considered independently of the others. As a result of this "half" step, the chemistry solver returns a new map of species concentrations over all cells, before transport occurs. The second step allows species to transport into/out of the cell, solving each species one at a time, but over all cells simultaneously. In this "half" step there is no further production or destruction of the species (that is, there are no chemical source terms in this sub-step); only transport terms are considered.

The key advantage to the operator splitting is that in the first "half-step", the solution of the net species production rates due to chemical reaction is achieved on a cell-by-cell basis. On this basis, we solve for all of the species simultaneously with temperature, which means that the equations are fully coupled. The second advantage is that we can apply a state-of-the-art ordinary-differential equation (ODE) solver to the task of integrating the stiff simultaneous equations within each cell. The operator-splitting approach allows each algorithm to do what it does best. The transport algorithm is applied to transport in time; the stiff kinetic solver is applied to net production/destruction due to kinetics and resulting change of species in a "closed" system with time. The "sub-cycling" or integration of the species equation over the specified transport time-step is done very efficiently using Ansys Forte's advanced, proprietary ODE solver that performs adaptive time-stepping with strict error control to assure a highly accurate solution in the most efficient time. The adaptive (sub) time steps are adjusted based on the largest rate of change for any species or temperature. The key advantage of this method is the adaptive selection of time step and sophistication in resolving both the trace and major species in the chemistry step. Furthermore, the proprietary ODE solution strategy takes advantage of the sparsity of the species-to-species reaction matrix in a typical combustion mechanism to further gain major efficiency benefits.

Ansys Forte also takes advantage of the operator-splitting method through implementation of parallel solution for the cell-by cell calculations on a parallel platform. In such cases, the chemistry calculation is easily load balanced over any number of processors.