Operator Splitting and Stiffness

Both explicit and implicit methods have many variations. One of the differences between these methods is the order of the polynomial that is used for the approximation of the solution. The more sophisticated methods provide more accurate solutions, but the most important is to use a temporal and/or spatial stepsize that allows the stability of the method (Higham 1996). 

In many situations, the method of operator splitting is applied to the solution of PDEs. In this case, the chemical kinetic step is separated from the transport steps and solved using ODE methods as described above. One of the advantages of operator splitting is that by separating the original convection-reaction-diffusion PDEs into different steps, it is possible to optimise solution methods that have been specifically developed for each submodel. Even if an implicit method has to be used for the chemical part, the matrices are far smaller than those resulting from the method of lines approach. 

Berkenbosch, A.C., Kaasschieter, E.F., Klein, R. Detonation capturing for stiff combustion chemistry. Combust. Theory Model. 2, 313-348 (1998) 

Berzins, M., Ware, J.M. Solving convection and convection-reaction problems using the method of lines. Appl. Numer. Math. 20, 83-99 (1996) 

Blasenbrey, T. Entwicklung und Implementierung automatisch reduzierter Reaktionsme-chanismen fiir die Verbrennung von Kohlenwasserstoffen. Stuttgart University (2000) Bongers, H., Van Oijen, J.A., De Goey, L.P.H. Intrinsic low-dimensional manifold method extended with diffusion. Proc. Combust. Inst. 29, 1371-1378 (2002) 

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