Showing papers in "Communications in Mathematical Sciences in 2004"
TL;DR: In this article, a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems is proposed and the convergence of the proposed method for nonlinear elliptic equations is studied.
Abstract: In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities.
222 citations
TL;DR: In this article, the authors derived new models for gravity driven shallow water flows in several space dimensions over a general topography, which are invariant under rotation, admit a conservative energy equation, and preserve the steady state of a lake at rest.
Abstract: We derive new models for gravity driven shallow water flows in several space dimensions over a general topography. A first model is valid for small slope variation, i.e. small curvature, and a second model is valid for arbitrary topography. In both cases no particular assumption is made on the velocity profile in the material layer. The models are written for an arbitrary coordinate system, and several formulations are provided. A Coulomb friction term is derived within the same framework, relevant in particular for debris avalanches. All our models are invariant under rotation, admit a conservative energy equation, and preserve the steady state of a lake at rest.
195 citations
TL;DR: In this article, a simplifled set of equations is derived systematically below for the interaction of large scale ∞ow flelds and precipitation in the tropical atmosphere, and the formal inflnitely fast relaxation limit converges to a novel hyperbolic free boundary problem.
Abstract: A simplifled set of equations is derived systematically below for the interaction of large scale ∞ow flelds and precipitation in the tropical atmosphere. These equations, the Tropical Climate Model, have the form of a shallow water equation and an equation for moisture coupled through a strongly nonlinear source term. This source term, the precipitation, is of relaxation type in one region of state space for the temperature and moisture, and vanishes identically elsewhere in the state space of these variables. In addition, the equations are coupled nonlinearly to the equations for barotropic incompressible ∞ow. Several mathematical features of this system are developed below including energy principles for solutions and their flrst derivatives independent of relaxation time. With these estimates, the formal inflnitely fast relaxation limit converges to a novel hyperbolic free boundary problem for the motion of precipitation fronts from a large scale dynamical perspective. Elementary exact solutions of the limiting dynamics involving precipitation fronts are developed below and include three families of waves: fast drying fronts as well as slow and fast moistening fronts. The last two families of waves violate Lax's Shock Inequalities; nevertheless, numerical experiments presented below conflrm their robust realizability with realistic flnite relaxation times. From the viewpoint of tropical atmospheric dynamics, the theory developed here provides a new perspective on the fashion in which the prominent large scale regions of moisture in the tropics associated with deep convection can move and interact with large scale dynamics in the quasi-equilibrium approximation.
179 citations
TL;DR: In this paper, a Petrov-Galerkin finite element method with nonconforming multiscale trial functions and linear test functions was proposed. And the authors showed that the cell resonance error is eliminated in this formulation and hence the convergence rate is greatly improved.
Abstract: We continue the study of the nonconforming multiscale finite element method (Ms- FEM) introduced in 17, 14 for second order elliptic equations with highly oscillatory coefficients. The main difficulty in MsFEM, as well as other numerical upscaling methods, is the scale resonance effect. It has been show that the leading order resonance error can be effectively removed by using an over-sampling technique. Nonetheless, there is still a secondary cell resonance error of O(Є^2/h^2). Here, we introduce a Petrov-Galerkin MsFEM formulation with nonconforming multiscale trial functions and linear test functions. We show that the cell resonance error is eliminated in this formulation and hence the convergence rate is greatly improved. Moreover, we show that a similar formulation can be used to enhance the convergence of an immersed-interface finite element method for elliptic interface problems.
118 citations
TL;DR: A conservative, second order accurate fully implicit discretization of ternary (three-phase) Cahn-Hilliard (CH) systems that has an associated discrete energy functional and a nonlinear multigrid method to efficiently solve the discrete system at the implicit time-level.
Abstract: 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 (13). 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 1 st order time step constraint for stability. We demonstrate convergence of our scheme numerically and we present several simulations of phase transitions in ternary systems.
95 citations
TL;DR: The SPDEs are derived by generalising the Langevin MCMC method to infinite dimensions by introducing a stochastic PDE based approach to sampling paths of SDEs, conditional on observations.
Abstract: We introduce a stochastic PDE based approach to sampling paths of SDEs, conditional on observations. The SPDEs are derived by generalising the Langevin MCMC method to infinite dimensions. Various applications are described, including sampling paths subject to two end-point conditions (bridges) and nonlinear filter/smoothers.
81 citations
TL;DR: In this article, the bifurcation and stability of the solutions of the Boussinesq equations and the onset of the Rayleigh-Benard convection were studied.
Abstract: We study in this article the bifurcation and stability of the solutions of the Boussinesq equations, and the onset of the Rayleigh-Benard convection. A nonlinear theory for this problem is established in this article using a new notion of bifurcation called attractor bifurcation and its corresponding theorem developed recently by the authors in [6]. This theory includes the following three aspects. First, the problem bifurcates from the trivial solution an attractor AR when the Rayleigh number R crosses the first critical Rayleigh number Rc for all physically sound boundary conditions, regardless of the multiplicity of the eigenvalue Rc for the linear problem. Second, the bifurcated attractor AR is asymptotically stable. Third, when the spatial dimension is two, the bifurcated solutions are also structurally stable and are classified as well. In addition, the technical method developed provides a recipe, which can be used for many other problems related to bifurcation and pattern formation.
70 citations
TL;DR: In this paper, the authors give a simplified proof of DiPerna's result for the relativistic version of the Vlasov-Maxwell system and show that the weak solutions preserve the total charge.
Abstract: In their seminal work (3), R. DiPerna and P.-L. Lions established the existence of global weak solutions to the Vlasov-Maxwell system. In the present notes we give a somewhat simplified proof of this result for the relativistic version of this system, the main purpose being to make this important result of kinetic theory more easily accessible to newcomers in the field. We show that the weak solutions preserve the total charge.
64 citations
TL;DR: In this article, the authors considered the damped and driven two-dimensional Navier-Stokes equations at the limit of small viscosity coefficiento! 0 +, and obtained upper bounds of the order o i 1 on the fractal and Hausdorff dimensions of the global attractor for the system on the torus T 2, on the sphere S 2 and in a bounded domain.
Abstract: We consider in this article the damped and driven two-dimensional Navier-Stokes equations at the limit of small viscosity coefficiento! 0 + . In particular, we obtain upper bounds of the order o i1 on the fractal and Hausdorff dimensions of the global attractor for the system on the torus T 2 , on the sphere S 2 and in a bounded domain. Furthermore, in the case of the torus, we establish a lower bound of the order o i1 . This sharp estimate is remarkably smaller than the well established sharp bound for the dimension of the global attractor of the Navier-Stokes equations on the torus T 2 , which is of the order o i4/3 . This means that the damping/friction term plays a significant role in reducing the number of degrees of freedom in this two-dimensional model. This, we believe, is done by dissipating the energy at the large spatial scales which is transferred to these scales via the inverse cascade mechanism. Finally, we remark that the system of equations studied here is related to the Stommel-Charney barotropic ocean circulation model of the gulf stream.
61 citations
TL;DR: In this article, the authors considered a central limit type problem, showing that p n(V n iV ) converges weakly, in the dual of a nuclear space, to the unique solution of a stochastic evolution equation.
Abstract: In an earlier paper, we studied the approximation of solutions V (t) to a class of SPDEs by the empirical measure V n (t) of a system of n interacting difiusions. In the present paper, we consider a central limit type problem, showing that p n(V n iV ) converges weakly, in the dual of a nuclear space, to the unique solution of a stochastic evolution equation. Analogous results in which the difiusions that determine V n are replaced by their Euler approximations are also discussed.
54 citations
TL;DR: In this paper, the authors study the dynamics of two-layer, stratified shallow water flows and find that unforced flows cannot reach the threshold of shear-instability, at least without breaking first.
Abstract: We study the dynamics of two-layer, stratified shallow water flows. This is a model in which two scenarios for eventual mixing of stratified flows (shear-instability and internal breaking waves) are, in principle, possible. We find that unforced flows cannot reach the threshold of shear-instability, at least without breaking first. This is a fully nonlinear stability result for a model of stratified, sheared flow. Mathematically, for 2X2 autonomous systems of mixed type, a criterium is found deciding whether the elliptic domain is reachable -smoothly- from hyperbolic initial conditions. If the characteristic fields depend smoothly on the system's Riemann invariants, then the elliptic domain is unattainable. Otherwise, there are hyperbolic initial conditions that will lead to incursions into the elliptic domain, and the development of the associated instability.
TL;DR: In this paper, a class of exact artificial boundary conditions for the numerical solution of the Schrodinger equation on unbounded domains in two-dimensional cases was proposed, based on the Fourier series expansion and the special functions techniques.
Abstract: In this paper, we propose a class of exact artificial boundary conditions for the numerical solution of the Schrodinger 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 greatly. Our numerical examples show that our method gives quite good numerical solutions with no numerical reflections.
TL;DR: In this article, a class of model prototype hybrid systems comprised of a microscopic stochastic surface process modeling adsorption/desorption and/or surface diusion of particles coupled to an ordinary di.erential equation (ODE) displaying bifurcations excited by a critical noise parameter.
Abstract: We introduce and study a class of model prototype hybrid systems comprised of a microscopic stochastic surface process modeling adsorption/desorption and/or surface di.usion of particles coupled to an ordinary di.erential equation (ODE) displaying bifurcations excited by a critical noise parameter. The models proposed here are caricatures of realistic systems arising in diverse applications ranging from surface processes and catalysis to atmospheric and oceanic models. We obtain deterministic mesoscopic models from the hybrid system by employing two methods: stochastic averaging principle and mean field closures. In this paper we focus on the case where phase transitions do not occur in the stochastic system. In the averaging principle case a faster stochastic mechanism is assumed compared to the ODE relaxation and a local equilibrium is induced with respect to the Gibbs measure on the lattice system. Under these circumstances remarkable agreement is observed between the hybrid system and the averaged system predictions. We exhibit several Monte Carlo simulations testing a variety of parameter regimes and displaying numerically the extent, limitations and validity of the theory. As expected fluctuation driven rare events do occur in several parameter regimes which could not possibly be captured by the deterministic averaging principle equation.
TL;DR: The main models presented are the Becker-Döring, fragmentationcoagulation (discrete or continuous) and Lifshitz-Slyozov ones, and some qualitative properties of the models, which include saturation, criticality, and dissipation are presented.
Abstract: This paper is meant as an introduction to some of the most classical models in the theory of fragmentation-coagulation. The main models presented are the Becker-Doring, fragmentation-coagulation (discrete or continuous) and Lifshitz-Slyozov ones. Rather than focusing on mathematical technicalities, we have chosen to insist on the physical ideas behind their derivation, in order to present them in a unified framework. The unifying physical principle in this context is the mass action principle, which we expose in detail, our philosophy being that these models may be thought of as technical variations on this theme. We then present some qualitative properties of the models, which include saturation, criticality, and dissipation. The second part of the paper collects some mathematical tools which are of recurrent use in this context, namely the use of moments, of the Laplace transform, and of Lyapunov functions.
TL;DR: In this paper, the authors apply the level set method to compute the three dimensional multivalued ge- ometrical optics term in a paraxial formulation, which is obtained from the 3D stationary eikonal equation by using one of the spatial directions as the evolution direction.
Abstract: We apply the level set method to compute the three dimensional multivalued ge- ometrical optics term in a paraxial formulation. The paraxial formulation is obtained from the 3-D stationary eikonal equation by using one of the spatial directions as the artiflcial evolution direction. The advection velocity fleld used to move level sets is obtained by the method of char- acteristics; therefore the motion of level sets is deflned in phase space. The multivalued travel-time and amplitude-related quantity are obtained from solving advection equations with source terms. We derive an amplitude formula in a reduced phase space which is very convenient to use in the level set framework. By using a semi-Lagrangian method in the paraxial formulation, the method has O(N 2 ) rather than O(N 4 ) memory storage requirement for up to O(N 2 ) multiple point sources in the flve dimensional phase space, where N is the number of mesh points along one direction. Although the computational complexity is still O(MN 4 ), where M is the number of steps in the ODE solver for the semi-Lagrangian scheme, this disadvantage is largely overcome by the fact that up to O(N 2 ) multiple point sources can be treated simultaneously. Three dimensional numerical examples demonstrate the e-ciency and accuracy of the method.
TL;DR: In this article, it was shown that the vanishing hyper-viscosity equation converges to the entropy solution of the corresponding convex conservation law under the Tartar-Murat compensated compactness theory.
Abstract: We prove that bounded solutions of the vanishing hyper-viscosity equation, ut + f(u)x +( 1) s "@ 2s x u = 0 converge to the entropy solution of the corresponding convex conservation law ut+f(u)x =0 ,f 00 > 0. The hyper-viscosity case, s> 1, lacks the monotonicity which underlines the Krushkov BV theory in the viscous case s = 1. Instead we show how to adapt the Tartar-Murat compensated compactness theory together with a weaker entropy dissipation bound to conclude the convergence of the vanishing hyper-viscosity.
TL;DR: In this paper, an adaptive mesh refinement algorithm based on the space-time staggered FDTD method is presented for local space-to-time mesh refinement appropriate for electromagnetic simulations based on electromagnetic simulations.
Abstract: An algorithm is presented for local space-time mesh refinement appropriate for electromagnetic simulations based on the space-time staggered FDTD method. The method is based on the adaptive mesh re.nement algorithm originally developed for hyperbolic conservation laws. Analysis of the dispersion relation and of the numerical reflection and transmission coefficients in one and two space dimensions shows that a scheme based on linear interpolation at the grid interfaces is unstable due to reflection coefficient > 1 at frequencies above the cutoff frequency of the coarse grid. A second-order accurate algorithm based on higher-order interpolations that enforces conservation of the magnetic field circulation at the fine-coarse grid boundaries is constructed. The new algorithm is shown to be stable and accurate for long time integration. A numerical simulation of an optical ring microcavity resonator using multilevel grid refinement in two space dimensions is presented.
TL;DR: In this paper, it was shown that the Nordstrom-Vlasov system converges to the Vlasov-Poisson system in a pointwise sense as the speed of light tends to infinity.
Abstract: The Nordstrom-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 Nordstrom-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.
TL;DR: In this paper, a front-tracking method for computing the moving contact line of two-dimensional drops and bubbles on a partially wetting surface exposed to shear flows is presented.
Abstract: In this paper we outline a front-tracking method for computing the moving contact line. In particular, we are interested in the motion of two-dimensional drops and bubbles on a partially wetting surface exposed to shear flows. Peskin’s Immersed Boundary Method is used to model the liquid-gas interface, similar to the approach used by Unverdi and Traggvason. The movement near the moving contact line is modelled by a slip condition, the value of the dynamic contact angle is determined by a linear model, and the local forces are introduced at the moving contact lines based on a relationship of moving contact angle and contact line speed. Numerical examples show that the method can be applied to the motion of drops and bubbles on a solid surface over a wide range of parameter values.
TL;DR: This work generalizes the functional analytical results of Meyer and applies them to a class of regression models, such as quantile, robust, logistic regression, for the analysis of multi- dimensional data.
Abstract: Recently Y. Meyer derived a characterization of the minimizer of the Rudin-Osher- Fatemi functional in a functional analytical framework. In statistics the discrete version of this functional is used to analyze one dimensional data and belongs to the class of nonparametric regres- sion models. In this work we generalize the functional analytical results of Meyer and apply them to a class of regression models, such as quantile, robust, logistic regression, for the analysis of multi- dimensional data. The characterization of Y. Meyer and our generalization is based on G-norm properties of the data and the minimizer. A geometric point of view of regression minimization is provided. whereDudenotes the total variation semi-norm of u and α> 0. The minimizer is called the bounded variation regularized solution. The taut-string algorithm consists in finding a string of minimal length in a tube (with radius α) around the primitive of f . The differentiated string is the taut-string reconstruction and corresponds to the minimizer of the ROF-model. Generalizing these ideas to higher dimensions is complicated by the fact that there is no obvious analog to primitives in higher space dimensions. We overcome this difficulty by solving Laplace's equation with right hand side f (i.e. integrate twice), and differentiating. The tube with radius α around the derivative of the potential specifies all functions u which satisfyu − fGs ≤ α (see also (21)). In this paper we show that the bounded variation regularized solutions (in any number of space dimensions) are contained in a tube of radius α .F or several other regression models in statistics, such as robust, quantile, and logistic regression (reformulated in a Banach space setting for analyzing multi-dimensional data) the
TL;DR: In this paper, divergence free vector fields have been shown to have nonuniqueness in the Cauchy problem for the ordinary differential equation defining characteristics, and there are smooth initial data so that the unique bounded solution is not continuous on any neighborhood of the origin.
Abstract: We give examples of divergence free vector fields. For such fields the Cauchy problem for the linear transport equation has unique bounded solutions. The first example has nonuniqueness in the Cauchy problem for the ordinary differential equation defining characteristics. In addition, there are smooth initial data so that the unique bounded solution is not continuous on any neighborhood of the origin. The second example is a field of similar regularity and intial data of bounded variation.
TL;DR: In this paper, it was shown that the orbital stability of solitary wave solutions of certain model equations is valid in L 2 -based Sobolev classes of arbitrarily high order, including the classical Korteweg-de Vries equation, the Benjamin-Ono equation and the cubic, nonlinear Schrodinger equation.
Abstract: 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 Schrodinger equation.
TL;DR: In this article, the authors present and analyze gradient recovery type a posteriori error estimates for the finite element approximation of elliptic variational inequalities of the second kind, and the reliability and efficiency of the estimates are addressed.
Abstract: 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.
TL;DR: In this paper, the authors derived macroscopic limits of the Becker-Döring equations for the special case of coagulation-fragmentation equations where clusters can gain or loose only one particle at a time.
Abstract: We review the derivation of macroscopic limits of the Becker-Döring equations. We show that those limits have the structure of a gradient flow even though the Becker-Döring equations themselves do not allow for such an interpretation. 1. The Becker–Döring equations The special case of coagulation-fragmentation equations where clusters can gain or loose only one particle at a time are known as the Becker-Döring equations. They were originally developed [3] to describe nucleation of liquid droplets in a supersaturated vapor. In the following we consider a homogeneous distribution of clusters which are characterized by their size l, the number of atoms in the cluster. We denote by cl(t) the concentration of l–clusters at time t. The net rate of conversion of l–clusters into (l+1)–clusters is denoted by Jl, which is measured in units of clusters per unit time per unit volume. The rate of change of the density of l–clusters is thus given by d dt cl(t) = Jl−1(t) − Jl(t) for l ≥ 2. (1.1) The density of free atoms is then determined by the constraint that the total number of atoms is conserved, i.e.
TL;DR: In this article, Baiti and Bressan showed that Glimm-type functionals can be extended to general functions with bounded variation and investigated their lower semi-continuity properties with respect to the strong L1 topology.
Abstract: Several Glimm-type functionals for (piecewise smooth) approximate solutions of nonlinear hyperbolic systems have been introduced in recent years. In this paper, following a work by Baiti and Bressan on genuinely nonlinear systems we provide a framework to prove that such functionals can be extended to general functions with bounded variation and we investigate their lower semi-continuity properties with respect to the strong L1topology. In particular, our result applies to the functionals introduced by Iguchi-LeFloch and Liu-Yang for systems with general flux-functions, as well as the functional introduced by Baiti-LeFloch-Piccoli for nonclassical entropy solutions. As an illustration of the use of continuous Glimm-type functionals, we also extend a result by Bressan and Colombo for genuinely nonlinear systems, and establish an estimate on the spreading of rarefaction waves in solutions of hyperbolic systems with general flux-function.
TL;DR: In this paper, the appearance of dust for a large class of deterministic and random fragmentations was investigated, and it was shown that a decrease of the total mass of the system due to the reduction into dust can be observed.
Abstract: For fragmentations in which particles split even faster when their mass is smaller, it is possible to observe a decrease of the total mass of the system, due to the reduction into dust. We investigate here this appearance of dust for a large class of deterministic and random fragmentations.
TL;DR: A multiscale method for a class of problems that are locally self-similar in scales and hence do not have scale separation, based on the framework of the HMM, to develop computational techniques that can extract accurately the macroscale behavior of the system under consideration, at a cost that is substantially less than the cost of solving the microscale problem.
Abstract: 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 self- similarity 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. 1. Introduction and HMM Upscaling In the last several years, there has been a tremendous growth of activity on developing multiscale methods in a number of application areas. For some reviews and perspectives, we refer to (4, 5, 8). The primary goal is to develop computational techniques that can extract accurately the macroscale behavior of the system under consideration, at a cost that is substantially less than the cost of solving the microscale problem. Obviously this can only be done if the problem under consideration has some special features that can be taken advantage of. Up to now, with the exception of renormalization group methods (1), all other existing multiscale techniques assume that there is scale separation in the problem, and this property is used in an essential way in order to design efficient multiscale methods (5, 8). While many problems, particularly problems with multi-physics, do have scale separation, there are other important problems with multiscales that do not have this property. The most well-known example is the fully developed turbulent flow whose active scales typically span continuously from the large energy pumping scale to the small energy dissipation scale, the ratio of these two scales was estimated by Kolomogorov to be of O(Re 3/4 )( 2, 6). HereRe is the Reynolds number of the flow which can easily reach 10 9 for atmospheric turbulence. In the absence of scale separation, we must seek other special features of these problems in order to develop efficient computational methods. The special feature that we will focus on in this paper is (statistical) self-similarity in scales, namely that the averaged properties of the system at different scales are related to each other by simple scaling factors. This feature is approximately satisfied by many important problems such as turbulent flows and sub-surface transport in the inertial range of scales. We will call these type D problems (for the definition of type A, type B and type C problems, see (3)). To develop an efficient computational technique, it is most convenient to use the framework of the heterogeneous multiscale method (HMM) (3). HMM has two main components: • Selecting the macroscale solver;
TL;DR: In this article, the authors present an algorithm to simulate the motion of rigid bodies subject to a non-overlapping constraint, and which tend to aggregate when they get close to each other.
Abstract: We present here an algorithm to simulate the motion of rigid bodies subject to a non–overlapping constraint, and which tend to aggregate when they get close to each other. The motion is induced by external forces. Two types of forces are considered here: drift force induced by the action of a surrounding fluid whose motion is prescribed, and stochastic forces modelling random shocks of molecules on the surface of the bodies. The numerical approach fits into the general framework of granular flow modelling.
TL;DR: In this paper, the concepts of white noise analysis are used to give an explicit solution to a stochastic transport equation driven by Levy white noise, and the authors demonstrate how to use white noise to solve a transport problem.
Abstract: In this paper we demonstrate how concepts of white noise analysis can be used to give an explicit solution to a stochastic transport equation driven by Levy white noise.