Numerical And Mathematical Analysis Of Some Combustion And Phase Separation Models

In this thesis,we will successively consider two free-interface problems:one is a thermo-diffusive combustion model,the other is a higher-order generalized Ca.hn-Hilliard equation.In both cases,we are interested in the stability analysis and nu-merical simulations.(I)Thermo-diffusive combustion:The problem models a thermo-diffusive combustion of premixed flame with zero-order reaction and stepwise temperature kinetics in a two-dimensional strip R x(-l/2,l/2),which reads:We restrict our analysis to cellular instabilities of the free interfaces,that is only to parameter regimes where the Lewis number is within 0<Le<1.To overcome the difficulty due to the presence of two interfaces(respectively the ignition interface F(t,y)and the trailing interface G(t,y)),we introduce in this framework a new method to study the stability of the fronts,based on the reduction of the free-interface problem to a fully nonlinear boundary value problem.Using a discrete Fourier transform,we compute explicitly the stability threshold,namely a critical value of Lewis number,Lc*,such that the planar traveling front is linearly asymptotically unstable for 0<Le<Lec*,stable for Le*<Le<1.It transpires that the number of admissible Fourier modes relies heavily on the strip width l.We complement our analysis with numerical simulations,to explore the insta-bility patterns produced by the model.Numerically,we observe that,after a rapid transition period,a steady configuration consisting of "two-cell" patterns for the ignition and trailing interfaces is established.(?)Higher-order Cahn-Hilliard equation:This type of equation reads:We devote ourself to some theoretical analysis under Dirichlet boundary condition-s,such as the well-posedness and regularities.We also prove the dissipativity of corresponding operators,as well as existence of a global attractor.On the one hand,we will give some numerical results for a higher-order Cahn-Hilliard-Oono equation,arsing from areas like biology and medicine,to be specific,a phase-field crystal equation coupled with a Cahn-Hilliard equation with a mass source,to simulate tumor growth.Our results show that anisotropy may be strongly influenced by the choice of coefficients in the higher-order terms.On the other hand,we consider a hyperbolic relaxation of the higer-order Cahn-Hilliard equation:We perform our numerical simulations with a second-order fully discrete scheme.The results also illustrate the effects of higher-order terms on the anisotropy.