Related papers: Accurate Spectral Numerical Schemes for Kinetic Eq…
We study the approximation and stability properties of a recently popularized discretization strategy for the speed variable in kinetic equations, based on pseudo spectral collocation on a grid defined by the zeros of a non-standard family…
Numerical techniques for discretization of velocity space in continuum kinetic calculations are described. An efficient spectral collocation method is developed for the speed coordinate - the radius in velocity space - employing a novel set…
We develop a spectrally accurate numerical method to compute solutions of a model partial differential equation used in plasma physics to describe diffusion in velocity space due to Fokker-Planck collisions. The solution is represented as a…
We address the problem of constructing accurate mathematical models of the dynamics of complex systems projected on a collective variable. To this aim we introduce a conceptually simple yet effective algorithm for estimating the parameters…
We consider two classes of linear kinetic equations: with constant collision frequency and constant mean free path of gas molecules (i.e., frequency of molecular collisions, proportional to the modulus molecular velocity). Based homogeneous…
This work presents a probabilistic scheme for solving semilinear nonlocal diffusion equations with volume constraints and integrable kernels. The nonlocal model of interest is defined by a time-dependent semilinear partial…
In this work we explore the fidelity of numerical approximations to the analytic spectra of hyperbolic partial differential equation systems with variable coefficients. We are particularly interested in the ability of discrete methods to…
An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete…
We construct numerical schemes to solve kinetic equations with anomalous diffusion scaling. When the equilibrium is heavy-tailed or when the collision frequency degenerates for small velocities, an appropriate scaling should be made and the…
Identification of nonlinear dynamical systems is crucial across various fields, facilitating tasks such as control, prediction, optimization, and fault detection. Many applications require methods capable of handling complex systems while…
Fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time. For processes lacking such scaling the corresponding description may be given by…
Physical laws governing population dynamics are generally expressed as differential equations. Research in recent decades has incorporated fractional-order (non-integer) derivatives into differential models of natural phenomena, such as…
High-precision numerical scheme for nonlinear hyperbolic evolution equations is proposed based on the spectral method. The detail discretization processes are discussed in case of one-dimensional Klein-Gordon equations. In conclusion, a…
Fractional Fokker-Planck equation plays an important role in describing anomalous dynamics. To the best of our knowledge, the existing discussions mainly focus on this kind of equation involving one diffusion operator. In this paper, we…
We present a class of new explicit and stable numerical algorithms to solve the spatially discretized linear heat or diffusion equation. After discretizing the space and the time variables like conventional finite difference methods, we do…
In a recent paper we presented a new ultra efficient numerical method for solving kinetic equations of the Boltzmann type (G. Dimarco, R. Loubere, Towards an ultra efficient kinetic scheme. Part I: basics on the 689 BGK equation, J. Comp.…
In this work, we develop novel structure-preserving numerical schemes for a class of nonlinear Fokker--Planck equations with nonlocal interactions. Such equations can cover many cases of importance, such as porous medium equations with…
Difference Kinetic Equations are derived quantum mechanically in a plane wavelets representation with account of two-particle correlations. It is shown that the set of plane wavelet orthonormal functions is complete. The set of ket vectors…
In this paper, we introduce and analyse numerical schemes for the homogeneous and the kinetic L\'evy-Fokker-Planck equation. The discretizations are designed to preserve the main features of the continuous model such as conservation of…
In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [10] for convection-diffusion equations, which relies on a…