Numerical simulation of reaction fronts in dissipative media

Numerical simulation of reaction fronts in dissipative media

Fronts of reaction in certain systems (such as so-called solid flames and detonation fronts) can be simulated by a single-equation phenomenological model of Strunin (1999, 2009). This is a high-order nonlinear partial differential equation describing the shape of the front as a function of spatial coordinates and time. The equation is of active-dissipative type, with 6th-order spatial derivative. For one-dimensional case, the equation was previously solved using the Galerkin method, but only one numerical experiment with limited information on the dynamics was obtained. For two-dimensional case only two numerical ex-
periments were reported so far, in which a low-accuracy infinite difference scheme was used. In this thesis, we use a more recent and sophisticated method, namely the one-dimensional integrated radial basis function networks (1D-IRBFN). The method had been developed by Tran-Cong and May-Duy (2001, 2003) and successfully applied to several problems such as structural analysis, viscoelastic flows and fluid-structure interaction. In contrast to commonly used approaches, where a function of interest is differentiated to give approximate derivatives, leading to a reduction in convergence rate for derivatives (and this reduction increases with derivative order, which magnifes errors), the 1D-IRBFN method uses the integral formulation. It utilizes spectral approximants to represent highest-order derivatives under consideration. They are then integrated analytically to yield approximate expressions for lower-order derivatives and the function itself.

In this thesis the following main results are obtained. A numerical program implementing the 1D-IRBFN method is developed in Matlab to solve the equation of interest. The program is tested by (a) constructing a forced version of the equation, which allows analytical solution, and verifying the numerical solution against the analytical solution; (b) reproducing one-dimensional spinning waves obtained from the model previously. A modified version of the program is successfully applied to similar high-order equations modelling auto-pulses in fluid flows with elastic walls.

We obtained numerically and analyzed a far richer variety of one-dimensional dynamics of the reaction fronts. Two kinds of boundary conditions were used: homogeneous conditions on the edges of the domain, and periodic conditions corresponding to periodicity of the front on a cylinder. The dependence of the dynamics on the size of the domain is explored showing how larger space accommodates multiple spinning waves. We determined the critical domain size (bifurcation point) at which non-trivial settled regimes become possible. We found a regime where the front is shaped as a pair of kinks separated by a relatively short distance. Interestingly, the pair moves in a stable joint formation far from the boundaries. A similar regime for three connected kinks is obtained. We demonstrated that the initial condition determines the direction of motion
of the kinks, but not their size and velocity. This is typical for active-dissipative systems. The settled character of these regimes is demonstrated. We also applied the 1D-IRBFN method to two-dimensional topology corresponding to a solid cylinder. Stable spinning wave solutions are obtained for this case.