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Communications in Mathematical Sciences
Communications in Mathematical Sciences
E, Weinan; Yue, Xing Y.
We present a multiscale method for a class of problems that are locally self-similar in scales and hence do not have scale separation. Our method is based on the framework of the heterogeneous multiscale method (HMM). At each point where macroscale data is needed, we perform several small scale simulations using the microscale model, then using the results and local selfsimilarity to predict the needed data at the scale of interest. We illustrate this idea by computing the effective macroscale transport of a percolation network at the percolation threshold.
Benedetto, Dario; Caglioti, Emanuele; Golse, François; Pulvirenti, Mario
This paper, which is a sequel to Benedetto-Caglioti-Golse-Pulvirenti, Comput. Math. Appl. 38 (1999), p. 121-131, considers as a starting point a mean-field equation for the dynamics of a gas of particles interacting via dissipative binary collisions. More precisely, we are concerned with the case where these particles are immersed in a thermal bath modeled by a linear Fokker-Planck operator. Two different scalings are considered for the resulting equation. One concerns the case of a thermal bath at finite temperature and leads formally to a nonlinear diffusion equation. The other concerns the case of a thermal bath at infinite temperature and leads formally to an isentropic...
Ben Abdallah, Naoufel; Chaker, Hédia
Han, Houde; Huang, Zhongyi
In this paper, we propose a class of exact artificial boundary conditions for the numerical solution of the Schrödinger equation on unbounded domains in two-dimensional cases. After we introduce a circular artificial boundary, we get an initial-boundary problem on a disc enclosed by the artificial boundary which is equivalent to the original problem. Based on the Fourier series expansion and the special functions techniques, we get the exact artificial boundary condition and a series of approximating artificial boundary conditions. When the potential function is independent of the radiant angle θ, the problem can be reduced to a series of one-dimensional problems. That can reduce the computational complexity...
Kim, Junseok; Kang, Kyungkeun; Lowengrub, John
We develop a conservative, second order accurate fully implicit discretization of ternary (three-phase) Cahn-Hilliard (CH) systems that has an associated discrete energy functional. This is an extension of our work for two-phase systems. We analyze and prove convergence of the scheme. To efficiently solve the discrete system at the implicit time-level, we use a nonlinear multigrid method. The resulting scheme is efficient, robust and there is at most a 1st order time step constraint for stability. We demonstrate convergence of our scheme numerically and we present several simulations of phase transitions in ternary systems.
Bona, Jerry L.; Liu, Yue; Nguyen, Nghiem V.
The orbital stability of solitary waves has generally been established in Sobolev classes of relatively low order, such as $H^1$. It is shown here that at least for solitary-wave solutions of certain model equations, a sharp form of orbital stability is valid in $L^2$-based Sobolev classes of arbitrarily high order. Our theory includes the classical Korteweg-de Vries equation, the Benjamin- Ono equation and the cubic, nonlinear Schrödinger equation.
Calogero, Simone; Lee, Hayoung
The Nordström-Vlasov system provides an interesting relativistic generalization of the Vlasov-Poisson system in the gravitational case, even though there is no direct physical application. The study of this model will probably lead to a better mathematical understanding of the class of non-linear systems consisting of hyperbolic and transport equations. In this paper it is shown that solutions of the Nordström-Vlasov system converge to solutions of the Vlasov-Poisson system in a pointwise sense as the speed of light tends to infinity, providing a further and rigorous justification of this model as a genuine relativistic generalization of the Vlasov-Poisson system.
Bostan, Viorel; Han, Weimin
In this paper, we present and analyze gradient recovery type a posteriori error estimates for the finite element approximation of elliptic variational inequalities of the second kind. Both reliability and efficiency of the estimates are addressed. Some numerical results are reported, showing the effectiveness of the error estimates in adaptive solution of elliptic variational inequalities of the second kind.
Cheng, Li-Tien; Liu, Hailiang; Osher, Stanley
We introduce a level set method for computational high frequency wave propagation in dispersive media and consider the application to linear Schrödinger equation with high frequency initial data. High frequency asymptotics of dispersive equations often lead to the well-known WKB system where the phase of the plane wave evolves according to a nonlinear Hamilton-Jacobi equation and the intensity is governed by a linear conservation law. From the Hamilton-Jacobi equation, wave fronts with multiple phases are constructed by solving a linear Liouville equation of a vector valued level set function in the phase space. The multi-valued phase itself can be constructed either from an additional linear hyperbolic equation...
Jin, Shi; Osher, Stanley
A three-factor interest rate model defined on a finite domain has been provided. All the functions in the model can be obtained from the real markets. It has been proven that a final-value problem of the corresponding partial differential equation on a finite domain has a unique solution. Because the formulation of the problem is on a finite domain and correct, it is not difficult to design efficient numerical methods for the problem. Therefore interest rate derivatives can be evaluated without any difficulty and the results can readily be used in practice.
Arnold, Anton; Ehrhardt, Matthias; Sofronov, Ivan
We propose a way to efficiently treat the well-known transparent boundary conditions for the Schrödinger equation. Our approach is based on two ideas: to write out a discrete transparent boundary condition (DTBC) using the Crank-Nicolson finite difference scheme for the governing equation, and to approximate the discrete convolution kernel of DTBC by sum-of-exponentials for a rapid recursive calculation of the convolution. ¶ We prove stability of the resulting initial-boundary value scheme, give error estimates for the considered approximation of the boundary condition, and illustrate the efficiency of the proposed method on several examples.
Minion, Michael L.
A semi-implicit formulation of the method of spectral deferred corrections (SISDC) for ordinary differential equations with both stiff and non-stiff terms is presented. Several modifications and variations to the original spectral deferred corrections method by Dutt, Greengard, and Rokhlin concerning the choice of integration points and the form of the correction iteration are presented. The stability and accuracy of the resulting ODE methods for both stiff and nonstiff problems are explored analytically and numerically. The SISDC methods are intended to be combined with the method of lines approach to yield a flexible framework for creating higher-order semi-implicit methods for partial differential equations. A discussion and numerical examples...
Groppi, Maria; Pennacchio, Micol
A class of reactive Euler-type equations derived from the kinetic theory of chemical reactions is presented and a finite-volume scheme for such problem is developed. The proposed method is based on a flux-vector splitting approach and it is second-order in space and time. The final non-linear problem coming from the discretization has a characteristic block diagonal structure that allows a decoupling in smaller subproblems. Finally, a set of numerical tests shows interesting behaviors in the evolution of the space-dependent fluid-dynamic fields driven by chemical reactions, not present in previous space homogeneous simulations.
Brenier, Yann; Mauser, Norbert; Puel, Marjolaine
We consider two different asymptotic limits of the Vlasov-Maxwell system describing a quasineutral plasma with a uniform ionic background. In the first case, as the magnetic field is preserved in the limiting process, we obtain the so-called electron magnetohydrodynamics equations. In the second case, we obtain the incompressible Euler equations with no more magnetic field left.
Dolbeault, Jean; Illner, Reinhard
In these notes we first introduce logarithmic entropy methods for time-dependent drift-diffusion equations and then consider a kinetic model of Vlasov-Fokker-Planck type for traffic flows. In the spatially homogeneous case the model reduces to a special type of nonlinear driftdiffusion equation which may permit the existence of several stationary states corresponding to the same density. Then we define general convex entropies and prove a convergence result for large times to steady states, even if more than one exists in the considered range of parameters, provided that some entropy estimates are uniformly bounded.
We study weak solutions to a combustion model problem. An equivalent conservation law with discontinuous flux is derived. Definition of an entropy solution is given, and the existence and uniqueness of the entropy solutions is proved. The convergence of a projection method and an implicit finite difference scheme is also proved. Finally using this approach we prove the convergence of a random projection method.
Le, Thinh; Du, Qiang
The MacPherson-Srolovitz formula has been recently established as a generalization of the two dimensional von Neumann relation for microstructure coarsening. In this paper, we present an extension of the MacPherson-Srolovitz formula under more general junction conditions.
Laurençot, Philippe; Perthame, Benoit
We consider the linear growth-fragmentation equation arising in the modelling of cell division or polymerisation processes. For constant coefficients, we prove that the dynamics converges to the steady state with an exponential rate. The control on the initial data uses an elaborate $L1$-norm that seems to be necessary. It also reflects the main idea of the proof, which is to use an anti-derivative of the solution. The main technical difficulty is related to the entropy dissipation rate, which is too weak to produce a Poincaré inequality.