Solution of Nonlinear Reaction-Diffusion Systems Based on Green’s Functions
Systematic methods and tools for managing the complexity
Advances in Computational & Numerical Methods (T4-4)
Keywords: Green’s functions, Integrators, Reaction diffusion models, Numerical solutions
Modern computer-based process design, optimization and control methodologies can require massive on-line solution of detailed, commonly distributed-parameter, models. For instance, optimization of chemical reactors with selectivity criteria requires the solution of the reaction-transport model for catalytic pellet and/or reactor scales. At each step of the optimization cycle the underlying reaction-transport model must be solved numerically by means of stable and robust schemes. Finite-differences and finite-elements schemes are widely used given the existence of both theoretical results on stability and computational techniques for implementation. In this way, it is apparent that the development of stable procedures for massive solution of distributed-parameter models seems to be a solved issue. Recent studies have shown that the usage of classical discretization techniques (e.g., orthogonal collocation, finite-differences, etc.) for reaction-diffusion models cannot be stable in a wide range of parameter values as required, for instance, in model parameter estimation. The main source of instability is the lower order compared to the domain for boundary conditions. Hence, the development of stable and robust numerical procedures for distributed parameter processes is still of up most importance within advanced process design and optimization methodologies.
From signal processing practice it is known that differentiators are, in fact, highly sensitive to round-off errors. Differentiation schemes are very likely to magnify the propagation of approximation errors and, hence, to reduce the accuracy of numerical solutions. Generally, this drawback is compensated by the usage of refined meshes. Since differential operators are analytically inverted within a Green's function formulation for distributed parameter processes, integral equation formulations for distributed parameter processes become a serious alternative to avoid the usage of approximate differentiators. In this approach, the differential equation is converted into an integral, Fredholm-type, equation where boundary conditions are incorporated exactly. As is known in signal processing and process control theories, integrators are welcome because of their ability to wash-out and smooth round-off errors. In this way, integral equation formulations offer the advantage that approximations for differentiators has not longer to be considered, and potential numerical schemes should rely only on numerical quadratures that are unconditionally stable.
In this paper, a further exploration of integral equation formulation for nonlinear and non-isothermal reaction-diffusion transport is carried out. To this end, the Green's function problem is posed and solved for the three geometries (i.e., rectangular, cylindrical and spherical), and two representative examples are worked out to illustrate the ability of the method to describe accurately the phenomena with respect to analytical and numerical solutions via finite-differences.
Presented Tuesday 18, 15:20 to 15:40, in session Advances in Computational & Numerical Methods (T4-4).
