Computational Singular Perturbation Theory

Lam and Goussis elaborated a detailed theory based on the application of computational perturbation methods for the investigation of reaction mechanisms. This family of methods is called computational singular perturbation theory and is often 

If the Jacobian of the kinetic system of differential equations has M eigenvalues with negative real parts that are much larger (i.e. more negative) than the other Re (Ay) values, then the solution is quickly attracted onto an (A/j-M)-dimensional surface Q, which is called the slow invariant manifold (SIM) (Fenichel 1979). Denote 7VJ2 and TyF as two subspaces, where the slow subspace TyFi is the space of movement on J2 and the fast subspace TyF contains the directions of fast approaches to the manifold. These spaces can be spanned by the following basis vectors TyF = span(a /= . M) and TyQ = span(a / = M +1. N ). Vectors a, form matrices A,. = [ai, a2. m and Aj = [Am+i, m+2 avJ- On this basis, the right-hand side of the kinetic ODE can be decomposed as 

Here ffast = ArZ and fsiow = AjZ, and the corresponding amplitudes are defined as 

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The boundary conditions follow naturally from the conventions Uq = = 0, and similarly for v, which say that there are no microorganisms in the two reservoirs. They are justified by the agreement between the numerically computed rest points of (5.1) and the solutions of (5.2) obtained by using singular perturbation theory (as in [S9]). 

Since boundary layers were first introduced by Prandtl at the start of the twentieth century, rapid strides have been made in the analytic and numerical investigation of such phenomena. It has also been realized that boundary and interior layer phenomena are ubiquitous in the problems of chemical physics. Nowhere have developments in this area been more notable than in the Russian school of singular perturbation theory and its application. The three chapters in this book are representative of the best analytic and computational work in this field in the second half of the century. 

However, it is more appropriate to provide theoretical justifications for such use. In this respect, first, we introduce the third category of decoupling of positive and negative states commonly known as the direct perturbation theory . This approach does not suffer from the singularity problems described previously. However, the four-component form of the Dirac equation remains intact. The new Hamiltonian requires identical computational effort as for the Dirac equation itself, hence it is not an attractive alternative to the Dirac equation. However, it is useful to assess the accuracy of approximate two-component forms derived from the Dirac equation such as Pauli Hamiltonian. Consider the transformation... 

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