Front Propagation Problems Research Articles

Variational estimates for the speed propagation of fronts in a nonlinear diffusive Fisher equation

We examine non-linear diffusive front propagation in the frame of the Fisher-type equation: ∂tu=∂xD(u)∂xu+u(1−u). We study the problem of a sudden jump in diffusivity motivated by models of glassy polymers. It is shown that this problem differs substantially from the problem of front propagation in the usual Fisher equation which was solved by Kolmogorov, Petrovsky, and Piskunov (KPP) in 1937. As in the Fisher, Kolmogorov, Petrovsky, Piskunov (FKPP) problem, the asymptotic dynamics of the non linear diffusive front propagation is reduced to the study of a nonlinear ordinary differential equation with adequate boundary conditions. Since this problem does not allow an exact result for the propagation speed, we use a variational approach to estimate the front speed and compare it with direct time-dependent numerical simulations showing an excellent agreement.

Quantitative stochastic homogenization of an unbounded front propagation problem

We obtain quantitative stochastic homogenization results for Hamilton–Jacobi equations arising in front propagation problems which move in the normal direction with a possible unbounded velocity. More precisely, we establish error estimates and rates of convergence for homogenization and effective Hamiltonian. The main idea is to perturb our unbounded problem by a bounded one, and to establish stability results in this context. Then, we combine the estimates that we find with the ones from the bounded case.

УСТОЙЧИВОСТЬ ФРОНТОВ ХИМИЧЕСКИХ ПРЕВРАЩЕНИЙ ВБЛИЗИ СОСТОЯНИЯ БЛОКИРОВАНИЯ

In the current work we consider a chemical reaction localized on the sharp interface between a solid and a diffusive constituent. The driving force for the reaction front propagation is a normal component of the chemical affinity tensor. The configuration of the deformed state where the driving force is zero corresponds to the reaction blocking state. The paper is aimed at studying this configuration. The utilized approach to analyze the stability of the equilibrium interphase between two deformable domains involves a linearized analysis of the kinetic equation for the perturbed interface. Previously this method was applied for an interface stability analysis in the case of phase transformations. The advantage of this method is that it also shows how the loss of stability occurs. An analytical solution for the perturbed kinetic equation is only possible for some simple configurations. Numerical procedures were applied in order to solve these types of problems. As an example, we used the numerical procedure, and the problem of the chemical reaction front propagation in a hollow cylinder is solved. For the unstable configuration we analyzed a stress state in the vicinity of the reaction blocking.

Front propagation and clustering in the stochastic nonlocal Fisher equation.

In this work, we study the problem of front propagation and pattern formation in the stochastic nonlocal Fisher equation. We find a crossover between two regimes: a steadily propagating regime for not too large interaction range and a stochastic punctuated spreading regime for larger ranges. We show that the former regime is well described by the heuristic approximation of the system by a deterministic system where the linear growth term is cut off below some critical density. This deterministic system is seen not only to give the right front velocity, but also predicts the onset of clustering for interaction kernels which give rise to stable uniform states, such as the Gaussian kernel, for sufficiently large cutoff. Above the critical cutoff, distinct clusters emerge behind the front. These same features are present in the stochastic model for sufficiently small carrying capacity. In the latter, punctuated spreading, regime, the population is concentrated on clusters, as in the infinite range case, which divide and separate as a result of the stochastic noise. Due to the finite interaction range, if a fragment at the edge of the population separates sufficiently far, it stabilizes as a new cluster, and the processes begins anew. The deterministic cutoff model does not have this spreading for large interaction ranges, attesting to its purely stochastic origins. We show that this mode of spreading has an exponentially small mean spreading velocity, decaying with the range of the interaction kernel.

Propagation of transition fronts in nonlinear chains with non-degenerate on-site potentials.

We address the problem of transition front propagation in chains with a bi-stable nondegenerate on-site potential and a nonlinear gradient coupling. For generic nonlinear coupling, one encounters a special regime of transitions, characterized by extremely narrow fronts, far supersonic velocities of the front propagation, and long waves in the oscillatory tail. This regime can be qualitatively associated with a shock wave. The front propagation can be described with the help of a simple reduced-order model; the latter delivers a kinetic law, which is almost not sensitive to the fine details of the on-site potential. Besides, it is possible to predict all main characteristics of the transition front, including its velocity, as well as the frequency and the amplitude of the oscillatory tail. Numerical results are in good agreement with the analytical predictions. The suggested approach allows one to consider the effects of an external pre-load, the next-nearest-neighbor coupling and the on-site damping. When the damping is moderate, it is possible to consider the shock propagation in the damped chain as a perturbation of the undamped dynamics. This approach yields reasonable predictions. When the damping is high, the transition front enters a completely different asymptotic regime of a subsonic kink. The gradient nonlinearity generically turns negligible, and the propagating front converges to the regime described by a simple exact solution for a continuous model with linear coupling.

Stochastic homogenization of a front propagation problem with unbounded velocity

We study the homogenization of Hamilton–Jacobi equations which arise in front propagation problems in stationary ergodic media. Our results are obtained for fronts moving with possible unbounded velocity. We show, by an example, that the homogenized Hamiltonian, which always exists, may be unbounded. In this context, we show convergence results if we start with a compact initial front. On the other hand, if the media satisfies a finite range of dependence condition, we prove that the effective Hamiltonian is bounded and obtain classical homogenization in this context.

High-order filtered scheme for front propagation problems

In this work we develop a specific application of the scheme proposed and analyzed in [1] to front propagation problems. The approach is based on the level-set method which leads in the isotropic case to a classical evolutive first order Hamilton- Jacobi equation.We will apply to this equation high-order “filtered schemes”, for these schemes the strong monotonicity property will not be satisfied. However, a weak є-monotonicity property applies and this is enough to obtain a convergence result and a precise error estimate. In the last section we will present several examples where we solve front propagation problems by filtered scheme in two and three dimensions showing the accuracy of our method.

Error estimates for approximation schemes of effective Hamiltonians arising in stochastic homogenization of Hamilton-Jacobi equations

We study approximation schemes for effective Hamiltonians arising in the homogenization of first order Hamilton-Jacobi equations in stationary ergodic settings. In particular, we prove error estimates concerning the rate of convergence of the approximated solution to the effective Hamiltonian. Our main motivations are front propagation problems, but our results can be generalized to other types of Hamiltonians.

Convergence of discontinuous Galerkin schemes for front propagation with obstacles

We study semi-Lagrangian discontinuous Galerkin (SLDG) and Runge-Kutta discontinuous Galerkin (RKDG) schemes for some front propagation problems in the presence of an obstacle term, modeled by a nonlinear Hamilton-Jacobi equation of the form $\min(u_t + c u_x, u - g(x))=0$, in one space dimension. New convergence results and error bounds are obtained for Lipschitz regular data. These "low regularity" assumptions are the natural ones for the solutions of the studied equations.

Anisotropic Fast Marching in ITK

The Fast Marching algorithm is an efficient numerical method for computing the distance and shortest path between points of a domain. For that purpose, it solves a front propagation problem, which can be of interest in itself. The method has numerous applications, ranging from motion planning to image segmentation. The unit of length, for computing the path length, may vary on the domain. Motivated by applications, we generalize the algorithm to the case where the unit of length also depends on the path direction. Segmentation methods can take advantage of this flexibility to achieve greater sensitivity and specificity, for a comparable computation time.

Front Propagation in Geometric and Phase Field Models of Stratified Media

We study front propagation problems for forced mean curvature flows and their phase field variants that take place in stratified media, i.e., heterogeneous media whose characteristics do not vary in one direction. We consider phase change fronts in infinite cylinders whose axis coincides with the symmetry axis of the medium. Using the recently developed variational approaches, we provide a convergence result relating asymptotic in time front propagation in the diffuse interface case to that in the sharp interface case, for suitably balanced nonlinearities of Allen-Cahn type. The result is established by using arguments in the spirit of $\Gamma$-convergence, to obtain a correspondence between the minimizers of an exponentially weighted Ginzburg-Landau type functional and the minimizers of an exponentially weighted area type functional. These minimizers yield the fastest traveling waves invading a given stable equilibrium in the respective models and determine the asymptotic propagation speeds for front-like initial data. We further show that generically these fronts are the exponentially stable global attractors for this kind of initial data and give sufficient conditions under which complete phase change occurs via the formation of the considered fronts.

A discontinuous Galerkin scheme for front propagation with obstacles

We are interested in front propagation problems in the presence of obstacles. We extend a previous work (Bokanowski et al. SIAM J Sci Comput 33(2):923–938, 2011), to propose a simple and direct discontinuous Galerkin (DG) method adapted to such front propagation problems. We follow the formulation of Bokanowski et al. (SIAM J Control Optim 48(7):4292–4316, (2010)), leading to a level set formulation driven by $$\min (u_t + H(x,\nabla u), u-g(x))=0$$ , where $$g(x)$$ is an obstacle function. The DG scheme is motivated by the variational formulation when the Hamiltonian $$H$$ is a linear function of $$\nabla u$$ , corresponding to linear convection problems in the presence of obstacles. The scheme is then generalized to nonlinear equations, written in an explicit form. Stability analysis is performed for the linear case with Euler forward, a Heun scheme and a Runge-Kutta third order time discretization using the technique proposed in Zhang and Shu (SIAM J Numer Anal 48:1038–1063, 2010). Several numerical examples are provided to demonstrate the robustness of the method. Finally, a narrow band approach is considered in order to reduce the computational cost.

Onsager approach to the one-dimensional solidification problem and its relation to the phase-field description

We give a general phenomenological description of the steady-state 1D front propagation problem in two cases: the solidification of a pure material and the isothermal solidification of two-component dilute alloys. The solidification of a pure material is controlled by the heat transport in the bulk and the interface kinetics. The isothermal solidification of two-component alloys is controlled by the diffusion in the bulk and the interface kinetics. We find that the condition of positive-definiteness of the symmetric Onsager matrix of interface kinetic coefficients still allows an arbitrary sign of the slope of the velocity-concentration line near the solidus in the alloy problem or of the velocity-temperature line in the case of solidification of a pure material. This result offers a very simple and elegant way to describe the interesting phenomenon of a possible non-single-value behavior of velocity versus concentration that has previously been discussed by different approaches. We also discuss the relation of this Onsager approach to the thin-interface limit of the phase-field description.

Short time uniqueness results for solutions of nonlocal and non-monotone geometric equations

We describe a method to show short time uniqueness results for viscosity solutions of general nonlocal and non-monotone second-order geometric equations arising in front propagation problems. Our method is based on some lower gradient bounds for the solution. These estimates are crucial to obtain regularity properties of the front, which allow to deal with nonlocal terms in the equations. Applications to short time uniqueness results for the initial value problems for dislocation type equations, asymptotic equations of a FitzHugh–Nagumo type system and equations depending on the Lebesgue measure of the fronts are presented.

A Discontinuous Galerkin Solver for Front Propagation

We propose a new discontinuous Galerkin method based on [Y. Cheng and C.-W. Shu, J. Comput. Phys., 223 (2007), pp. 398–415] to solve a class of Hamilton–Jacobi equations that arises from optimal control problems. These equations are connected to front propagation problems or minimal time problems with nonisotropic dynamics. Several numerical experiments show the relevance of our method, in particular, for front propagation.

Front propagation problems with sub-diffusion

Front propagation is considered for two kinds of sub-diffusion - reaction systems: (i) systems with sub-diffusion limited reaction rate governed by models with fractional time derivatives; (ii) systems with activation limited reaction rate governed by integro-differential equations with two time variables. It is shown that in the former case the front is described by a travelling wave solution, while in the latter the velocity of the front decreases with time.

Exact solutions in front propagation problems with superdiffusion

Front propagation in a number of reaction–superdiffusion problems is studied. Specifically, traveling wave propagation, domain wall pinning, and systems of waves governed by a bistable single equation as well as FitzHugh–Nagumo equations are considered. The reaction terms in the equations are taken in the form of piecewise linear functions, which allow for exact solutions to be obtained. The effect of superdiffusion on front propagation is discussed.

An Efficient Data Structure and Accurate Scheme to Solve Front Propagation Problems

In this paper, we are interested in some front propagation problems coming from control problems in d-dimensional spaces, with d?2. As opposed to the usual level set method, we localize the front as a discontinuity of a characteristic function. The evolution of the front is computed by solving an Hamilton-Jacobi-Bellman equation with discontinuous data, discretized by means of the antidissipative Ultra Bee scheme. We develop an efficient dynamic storage technique suitable for handling front evolutions in large dimension. Then we propose a fast algorithm, showing its relevance on several challenging tests in dimension d=2,3,4. We also compare our method with the techniques usually used in level set methods. Our approach leads to a computational cost as well as a memory allocation scaling as O(N nb ) in most situations, where N nb is the number of grid nodes around the front. Moreover, we show on several examples the accuracy of our approach when compared with level set methods.

Minimum time control problems for non-autonomous differential equations

In this paper, we investigate a minimum time problem for controlled non-autonomous differential systems, with dynamics depending on the final time. The minimal time function associated to this problem does not satisfy the dynamic programming principle. However, we will prove that it is related to a standard front propagation problem via the reachability function. Two simple numerical examples are given to illustrate our approach.

Some Convergence Results for Howard's Algorithm

This paper deals with convergence results of Howard's algorithm for the resolution of $\min_{a\in\mathcal{A}} (B^a x - c^a) = 0$, where $B^a$ is a matrix, $c^a$ is a vector, and $\mathcal{A}$ is a compact set. We show a global superlinear convergence result, under a monotonicity assumption on the matrices $B^a$. Extensions of Howard's algorithm for a max-min problem of the form $\max_{b\in\mathcal{B}} \min_{a\in\mathcal{A}} (B^{a,b}x - c^{a,b}) = 0$ are also proposed. In the particular case of an obstacle problem of the form $\min (Ax - b,$ $x - g) = 0$, where A is an $N \times N$ matrix satisfying a monotonicity assumption, we show the convergence of Howard's algorithm in no more than N iterations, instead of the usual $2^N$ bound. Still in the case of an obstacle problem, we establish the equivalence between Howard's algorithm and a primal-dual active set algorithm [M. Hintermüller, K. Ito, and K. Kunisch, SIAM J. Optim., 13 (2002), pp. 865–888]. The algorithms are illustrated on the discretization of nonlinear PDEs arising in the context of mathematical finance (American option and Merton's portfolio problem), of front propagation problems, and for the double-obstacle problem.