Time Semi-discrete Scheme Research Articles

A splitting scheme for the quantum Liouville-BGK equation

We introduce in this work an efficient numerical method for the simulation of the quantum Liouville-BGK equation, which models the diffusive transport of quantum particles. The corner stone to the model is the BGK collision operator, obtained by minimizing the quantum free energy under the constraint that the local density of particles is conserved during collisions. This leads to a large system of coupled nonlinear nonlocal PDEs whose resolution is challenging. We then define a splitting scheme that separates the transport and the collision parts, which, exploiting the local conservation of particles, leads to a fully linear collision step. The latter involves the resolution of a constrained optimization problem that is handled with the nonlinear conjugate gradient algorithm. We prove that the time semi-discrete scheme is convergent, and as an application of our numerical scheme, we validate the quantum drift-diffusion model that is obtained as the diffusive limit of the quantum Liouville-BGK equation.

A localized meshless technique for solving 2D nonlinear integro-differential equation with multi-term kernels

This paper studies an accurate localized meshless collocation approach for solving two-dimensional nonlinear integro-differential equation (2D-NIDE) with multi-term kernels. The proposed strategy discretizes the unknown solution in two phases. First, the semi-discrete scheme is obtained by using backward Euler finite difference (FD) approach and the first-order convolution quadrature rule for the first order temporal derivative and the Riemann-Liouville (R-L) fractional integral, respectively. Second, the spatial discretization is established by means of the local radial basis function based on partition of unity (LRBF-PU) in the space variable and its partial derivatives. Furthermore, the unconditionally stable result and first-order convergence of the time semi-discrete scheme in L2-norm are proved by the energy method. It is shown that the proposed method is accurate and that the numerical results support the theoretical analysis.

Analysis of an asymptotic preserving low mach number accurate IMEX-RK scheme for the wave equation system

In this paper the analysis of an asymptotic preserving (AP) IMEX-RK finite volume scheme for the wave equation system in the zero Mach number limit is presented. An IMEX-RK methodology is employed to obtain a time semi-discrete scheme, and a space-time fully-discrete scheme is derived by using standard finite volume techniques. The existence of a unique numerical solution, its uniform stability with respect to the Mach number, and the accuracy at low Mach numbers are established for both time semi-discrete and space-time fully-discrete schemes. The AP property of the scheme is proved for a general class of IMEX schemes which need not be globally stiffly accurate. Extensive numerical case studies confirm uniform second order convergence of the scheme with respect to the Mach number and all the above-mentioned properties.

Drift coefficient inversion problem of Kolmogorov-type equation

Kolmogorov-type equations often appear in stochastic analysis and have important applications in financial derivatives pricing, stochastic control and other fields. In this paper, we consider an inverse problem of reconstructing drift coefficient in a Kolmogorov-type equation. Being different from other works, the unknown drift coefficient is related to both temporal and spatial variables, which makes theoretical analysis rather difficult. Until now, documents dealt with evolutional inverse drift problems are quite few. Inspired by the Rothe's idea, we introduce a new time semi-discrete scheme to find the optimal solution at each time layer. Then we construct an approximate solution of the unknown drift coefficient and strictly analyze its convergence. After establishing the necessary conditions for the limit minimizer, we prove the uniqueness and stability of the global optimal solution.

Application of spectral element method for solving Sobolev equations with error estimation

This paper is dedicated to numerically solving the Sobolev equations that have several applications in physics and mechanical engineering. First, the temporal derivative is discretized by the Crank-Nicolson finite difference technique to obtain a semi-discrete scheme in the temporal direction. Afterward, the stability and convergence analysis of the time semi-discrete scheme are proven by applying the energy method. It also implies that the convergence order in the temporal direction is O(dt2). Second, a fully discrete formula has been acquired by discretizing the spatial derivatives via Legendre spectral element method (LSEM). This method applies the Lagrange polynomial based on the Gauss-Legendre-Lobatto (GLL) points. Moreover, an error estimation is given for the obtained fully discrete scheme. Eventually, the two-dimensional Sobolev equations are solved by using the proposed procedure. The accuracy and efficiency of the mentioned procedure are demonstrated by several numerical examples.

Numerical evaluation of fractional Tricomi-type model arising from physical problems of gas dynamics

This paper deals with approximating the time fractional Tricomi-type model in the sense of the Caputo derivative. The model is often adopted for describing the anomalous process of nearly sonic speed gas dynamics. The temporal semi-discretization is computed via a finite difference algorithm, while the spatial discretization is obtained using the local radial basis function in a finite difference mode. The local collocation method approximates the differential operators using a weighted sum of the function values over a local collection of nodes (named stencil) through a radial basis function expansion. This technique considers merely the discretization nodes of each subdomain around the collocation node. This leads to sparse systems and tackles the ill-conditioning produced of global collocation. The theoretical convergence and stability analyses of the proposed time semi-discrete scheme are proved by means of the discrete energy method. Numerical results confirm the accuracy and efficiency of the new approach.

A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions

<p style="text-indent:20px;">We propose a time semi-discrete scheme for the Caginalp phase-field system with singular potentials and dynamic boundary conditions. The scheme is based on a time splitting which decouples the equations and on a convex splitting of the energy associated to the problem. The scheme is unconditionally uniquely solvable and the energy is nonincreasing if the time step is small enough. The discrete solution is shown to converge to the energy solution of the problem as the time step tends to <inline-formula><tex-math id="M1">\begin{document}$ 0 $\end{document}</tex-math></inline-formula>. The proof involves a multivalued operator and a monotonicity argument. This approach allows us to compute numerically singular solutions to the problem.

A local meshless method to approximate the time-fractional telegraph equation

In the present work, we investigate the numerical solution of time-fractional telegraph equation by a local meshless method. The fractional-order derivative is defined in the Caputo’s sense. The time semi-discretization was carried out using finite difference method followed by radial basis function-based spatial discretization. The theoretical convergence analysis and stability analysis of time semi-discrete scheme are also proved. Several test problems with regular and irregular domains with uniform and non-uniform points are considered. To demonstrate the accuracy and efficiency of the proposed method, we compared the analytical and numerical solution of the proposed problem.

Numerical methods for the time fractional diffusion equation

In this paper, a differential analog of the Caputo fractional derivative called the L2 − 1σ formula is considered. The time semi-discrete scheme of the time fractional diffusion equation is obtained by using the L2 − 1σ formula to discretize the time derivative. The Sinc-Collocation method and the Sinc-Galerkin method are used to discretize the spatial derivatives, and the numerical schemes of the time fractional diffusion equation are constructed. Finally, the validity of the numerical schemes of this paper is verified by numerical example.

A meshless local collocation method for time fractional diffusion wave equation

In this manuscript, we present a radial basis function based local collocation method for solving time fractional diffusion-wave equation. The advantage of the local collocation method is only the discretization points located in each sub-domain, surrounding the collocation point, need to be considered. It also avoids the ill-conditioning problem arises due to the use of global collocation method. The theoretical stability and convergence of the proposed time semi-discrete scheme are also discussed. Numerical experiments for one dimension and two dimension problems are carried out. It is shown that the present method is efficient and numerical results have nice agreements with theoretical result.

An explicit spectral collocation method for the linearized Korteweg–de Vries equation on unbounded domain

In this paper, we present a stable and efficient numerical scheme for the linearized Korteweg–de Vries equation on unbounded domain. After employing the Crank–Nicolson method for temporal discretization, the transparent boundary conditions are derived for the time semi-discrete scheme. Then the unconditional stability of the resulting initial boundary problem is established. For spatial discretization, we construct a non-polynomial based spectral collocation method in which the basis functions are built upon a generalized Birkhoff interpolation. The interpolation error of the new basis is also investigated. Moreover, the basis functions build in two free parameters intrinsically which can be chosen properly so that the implicit time semi-discrete scheme collapses to an explicit scheme after spatial discretization. Numerical tests are performed to demonstrate the stability and accuracy of the proposed method.

A semi-discrete scheme for the stochastic Landau–Lifshitz equation

We propose a new convergent time semi-discrete scheme for the stochastic Landau-Lifshitz-Gilbert equation. The scheme is only linearly implicit and does not require the resolution of a nonlinear problem at each time step. Using a martingale approach, we prove the convergence in law of the scheme up to a subsequence.

Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation

AbstractWe present the finite difference/element method for a two-dimensional modified fractional diffusion equation. The analysis is carried out first for the time semi-discrete scheme, and then for the full discrete scheme. The time discretization is based on theL1-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term. We use finite element method for the spatial approximation in full discrete scheme. We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent. Moreover, the optimal convergence rate is obtained. Finally, some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.

Two regularization strategies for an evolutional type inverse heat source problem

This paper investigates an evolutional type inverse problem of determining an unknown heat source function in heat conduction equations when the solution is known in a discrete point set. Being different from other ordinary inverse source problems which often rely on only one variable, the unknown coefficient in this paper depends not only on the space variable x, but also on time t. Two regularization strategies which are called the time semi-discrete scheme (TSDS) and the integral reconstruction scheme (IRS), respectively, are proposed to deal with such a problem. By the TSDS the inverse problem is transformed into a sequence of stationary inverse problems and the unknown heat source is reconstructed layer by layer, while the IRS is to recover the source function from the situation as a whole. Both theoretical and numerical studies are provided. Two numerical algorithms on the basis of the Landweber iteration are designed, and some typical numerical experiments are performed in this paper. The numerical results show that the proposed methods are stable and the unknown heat source is recovered very well.

Optimization method for an evolutional type inverse heat conduction problem

This paper deals with the determination of a pair (q, u) in the heat conduction equation with initial and boundary conditions from the overspecified data u(x, t) = g(x, t). By the time semi-discrete scheme, the problem is transformed into a sequence of inverse problems in which the unknown coefficients are purely space dependent. Based on the optimal control framework, the existence, uniqueness and stability of the solution (q, u) are proved. A necessary condition which is a couple system of a parabolic equation and parabolic variational inequality is deduced.

Moving mesh for the axisymmetric harmonic map flow

We build corotational symmetric solutions to the harmonic map flow from the unit disc into the unit sphere which have constant degree. First, we prove the existence of such solutions, using a time semi-discrete scheme based on the idea that the harmonic map flow is the L 2 -gradient of the relaxed Dirichlet energy. We prove a partial uniqueness result concerning these solutions. Then, we compute numerically these solutions by a moving-mesh method which allows us to deal with the singularity at the origin. We show numerical evidence of the convergence of the method. Mathematics Subject Classification. 35A05, 35K55, 65N30, 65N50, 65N99.