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In this paper, we consider the Galerkin finite element method for the two-dimensional space and time fractional Bloch–Torrey equation (2D-STFBTE). Utilizing the L 2 − 1 σ formula to discretize temporal Caputo derivative, we obtain the local truncation error with O ( τ 3 − α ) in the temporal direction, where α is the order of time fractional derivative. Furthermore, the semi-discrete form for the problem is given and, the stability and convergence of the semi-discrete variational formulation are rigorously proved in L 2 norm and fractional norm respectively. Then, we derive a fully discrete scheme for the 2D-STFBTE and investigate its convergence theoretically. Finally, extensive numerical results based on linear piecewise polynomials verify the theoretical results and show the effectiveness of the proposed scheme.

In this paper, we study a V-cycle multigrid method for linear systems arising from time-dependent two-dimensional space-fractional diffusion equations. The coefficient matrices of the linear systems are structured such that their matrix-vector multiplications can be computed efficiently. The main advantage using the multigrid method is to handle the space-fractional diffusion equations on non-rectangular domains, and to solve the linear systems with non-constant coefficients more effectively. The main idea of the proposed multigrid method is to employ two banded splitting iteration schemes as pre-smoother and post-smoother. The pre-smoother and the post-smoother take banded splitting of the coefficient matrix under the x-dominant ordering and the y-dominant ordering, respectively. We prove the convergence of the proposed two banded splitting iteration schemes in the sense of infinity norm. Results of numerical experiments for time-dependent two-dimensional space-fractional diffusion equations on rectangular, L-shape and U-shape domains are reported to demonstrate that both computational time and iteration number required by the proposed method are significantly smaller than those of the other tested methods.

A numerical method for a kind of time-dependent two-dimensional two-sided space fractional diffusion equations is developed in this paper. The proposed method combines a time scheme based on direction splitting approaches and a spectral method for the spatial discretization. The direction splitting approach renders the underlying two-dimensional equation into a set of one-dimensional space fractional diffusion equations at each time step. Then these one-dimensional equations are solved by using the spectral method based on weak formulations. A time error estimate is derived for the semi-discrete solution, and the unconditional stability of the fully discretized scheme is proved. Some numerical examples are presented to validate the proposed method.

A new model for simulating contact line dynamics is proposed. We apply the idea of driving contact-line movement by enforcing the equilibrium contact angle at the boundary, to the conservative level set method for incompressible two-phase flow [E. Olsson, G. Kreiss, A conservative level set method for two phase flow, J. Comput. Phys. 210 (2005) 225-246]. A modified reinitialization procedure provides a diffusive mechanism for contact-line movement, and results in a smooth transition of the interface near the contact line without explicit reconstruction of the interface. We are able to capture contact-line movement without loosing the conservation. Numerical simulations of capillary dominated flows in two space dimensions demonstrate that the model is able to capture contact line dynamics qualitatively correct.

The development of efficient algorithms to analyze complex electromagnetic structures is of topical interest. Application of these algorithms in commercial solvers requires rigorous error controllability. In this paper we focus on the perfectly matched layer based multilevel fast multipole algorithm (PML-MLFMA), a dedicated technique constructed to efficiently analyze large planar structures. More specifically the crux of the algorithm, viz. the pertinent layered medium Green functions, is under investigation. Therefore, particular attention is paid to the plane wave decomposition for 2-D homogeneous space Green functions in very lossy media, as needed in the PML-MLFMA. The result of the investigations is twofold. First, upper bounds expressing the required number of samples in the plane wave decomposition as a function of a preset accuracy are rigorously derived. These formulas can be used in 2-D homogeneous (lossy) media MLFMAs. Second, a more heuristic approach to control the error of the PML-MLFMA's Green functions is presented. The theory is verified by means of several numerical experiments.

This paper describes a new numerical scheme for the approximation of steady state solutions to systems of hyperbolic conservation laws. It generalises the fluctuation distribution framework by allowing the underlying representation of the solution to be discontinuous. This leads to edge-based fluctuations in addition to the standard cell-based fluctuations, which are then distributed to the cell vertices in an upwind manner which retains the properties of the continuous scheme (positivity, linearity preservation, conservation, compactness and continuity). Numerical results are presented on unstructured triangular meshes in two space dimensions for linear and nonlinear scalar equations as well as the Euler equations of gasdynamics. The accuracy of the approximation in smooth regions of the flow is shown to be very similar to the corresponding continuous scheme, but the discontinuous approach improves the sharpness with which discontinuities in the flow can be captured and provides additional flexibility which will allow adaptive techniques to be applied simply to improve efficiency.

A new set of consistent boundary conditions for Yee scheme approximations of wave equations in two space dimensions are developed and analyzed. We show how the classical staircase boundary conditions for hard reflections or, in the electromagnetic case, conducting surfaces in certain cases give O(1) errors. The proposed conditions keep the structure of the Yee scheme and are thus well suited for high performance computing. The higher accuracy is achieved by modifying the coefficients in the difference stencils near the boundary. This generalizes our earlier results with Gustafsson and Wahlund in one space dimension. We study stability and convergence and we present numerical examples.

In this paper, the second one in the series beginning with D. Mao, Towards front tracking based on conservation in two space dimensions, SIAM J. Sci. Comput. 22 (1) (2000) 113-151], we study another important feature of our 2D conservative front-tracking method, i.e. discontinuity curves in two space dimensions are tracked in a 1D capturing fashion. The evolution of 2D discontinuity curves are locally described by 1D conservation laws with source terms, which are derived from the governing equations. The front-tracking in our method is then realized by numerically simulating these 1D conservation laws with source terms in a conservative fashion. In this paper, our 2D front-tracking method is described in details, which is Cartesian-grid-based, conservative and much simpler in algorithm than other 2D front-tracking methods. The discussion starts with the 1D case, which facilitates the following 2D discussion. Data structure of the numerical solutions and first- and second-order versions of our 2D front-tracking method are described. Finally, numerical examples for both scalar equations and the Euler system of gas dynamics in 2D are presented to show the efficiency and effectiveness of the method.

In this paper, we propose a pseudo-spectral multiscale method for simulating complex systems with more than one spatial scale. Using a spectral decomposition, we split the displacement into its mean and fluctuation parts. Under the assumption of localized nonlinear fluctuations, we separate the domain into an MD (Molecular Dynamics) subdomain and an MC (MacrosCopic) subdomain. An interfacial condition is proposed across the two scales, in terms of a time history treatment. In the special case of a linear system, this is the first exact interfacial condition for multiscale computations. Meanwhile, we design coarse grid equations using a matching differential operator approach. The coarse grid discretization is of spectral accuracy. We do not use a handshaking region in this method. Instead, we define a coarse grid over the whole domain and reassign the coarse grid displacement in the MD subdomain with an average of the MD solution. To reduce the computational cost, we compute the dynamics of the coarse grid displacement and relate it to the mean displacement. Our method is therefore called a pseudo-spectral multiscale method. It allows one to reach high resolution by balancing the accuracy at the coarse grid with that at the interface. Numerical experiments in one- and two-space dimensions are presented to demonstrate the accuracy and the robustness of the method.

In this paper, we first briefly review the semi-analytical method [E.F. Toro, V.A. Titarev, Solution of the generalized Riemann problem for advection-reaction equations, Proc. Roy. Soc. London 458 (2018) (2002) 271-281] for solving the derivative Riemann problem for systems of hyperbolic conservation laws with source terms. Next, we generalize it to hyperbolic systems for which the Riemann problem solution is not available. As an application example we implement the new derivative Riemann solver in the high-order finite-volume ADER advection schemes. We provide numerical examples for the compressible Euler equations in two space dimensions which illustrate robustness and high accuracy of the resulting schemes.

The maximum entropy principle is a versatile tool for evaluating smooth approximations of probability density functions with a least bias beyond given constraints. In particular, the moment-based constraints are often a common prior information about a statistical state in various areas of science, including that of a forecast ensemble or a climate in atmospheric science. With that in mind, here we present a unified computational framework for an arbitrary number of phase space dimensions and moment constraints for both Shannon and relative entropies, together with a practical usable convex optimization algorithm based on the Newton method with an additional preconditioning and robust numerical integration routine. This optimization algorithm has already been used in three studies of predictability, and so far was found to be capable of producing reliable results in one- and two-dimensional phase spaces with moment constraints of up to order 4. The current work extensively references those earlier studies as practical examples of the applicability of the algorithm developed below.

It is shown that a discrete delta function can be constructed using a technique developed by Anita Mayo [The fast solution of Poisson's and the biharmonic equations on irregular regions, SIAM J. Sci. Comput. 21 (1984) 285-299] for the numerical solution of elliptic equations with discontinuous source terms. This delta function is concentrated on the zero level set of a continuous function. In two space dimensions, this corresponds to a line and a surface in three space dimensions. Delta functions that are first and second order accurate are formulated in both two and three dimensions in terms of a level set function. The numerical implementation of these delta functions achieves the expected order of accuracy.

We have constructed a lattice Wigner-Weyl code to expand the Buot-Jensen algorithm to calculation of electron transport in two-dimensional cylindrically symmetric structures. Almost all of the numerical simulations to date have dealt with the restricted problem of one-dimensional transport. In real devices, electrons are not confined to a single transport dimension and the coulombic potential is fully present and felt in three dimensions. We show the derivation of the 2D equation in cylindrical coordinates as well as approximations employed in the calculation of the four-dimensional convolution integral of the Wigner function and the potential. We work under the assumption that longitudinal transport is more dominant than radial transport and employ parallel processing techniques. The total transport is calculated in two steps: (1) transport the particles in the longitudinal direction in each shell separately, then (2) each shell exchanges particles with its nearest neighbor. Most of this work is concerned with the former step: A 1D space and 2D momentum transport problem. Time evolution simulations based on these method are presented for three different cases. Each case lead to numerical results consistent with expectations. Discussions of future improvements are discussed.

In this paper we propose a new finite volume evolution Galerkin (FVEG) scheme for the shallow water magnetohydrodynamic (SMHD) equations. We apply the exact integral equations already used in our earlier publications to the SMHD system. Then, we approximate these integral equation in a general way which does not exploit any particular property of the SMHD equations and should thus be applicable to arbitrary systems of hyperbolic conservation laws in two space dimensions. In particular, we investigate more deeply the approximation of the spatial derivatives which appear in the integral equations. The divergence free condition is satisfied discretely, i.e. at each vertex. First numerical results confirm reliability of the numerical scheme.

In this paper, we generalize a technique of anti-diffusive flux corrections, recently introduced by Despres and Lagoutiere [Journal of Scientific Computing 16 (2001) 479-524] for first-order schemes, to high order finite difference weighted essentially non-oscillatory (WENO) schemes. The objective is to obtain sharp resolution for contact discontinuities, close to the quality of discrete traveling waves which do not smear progressively for longer time, while maintaining high order accuracy in smooth regions and non-oscillatory property for discontinuities. Numerical examples for one and two space dimensional scalar problems and systems demonstrate the good quality of this flux correction. High order accuracy is maintained and contact discontinuities are sharpened significantly compared with the original WENO schemes on the same meshes.

In a previous paper, it was shown that consistent high-order discretization can make a Lax-Wendroff method stable in the presence of small cells near the boundary in one space and one time dimension. Here I show that a straightforward extension of this approach to two space dimensions is similarly successful.

We introduce two new volume-of-fluid interface reconstruction algorithms and compare the accuracy of these algorithms to four other widely used volume-of-fluid interface reconstruction algorithms. We find that when the interface is smooth (e.g., continuous with two continuous derivatives) the new methods are second-order accurate and the other algorithms are first-order accurate. We propose a design criteria for a volume-of-fluid interface reconstruction algorithm to be second-order accurate. Namely, that it reproduce lines in two space dimensions or planes in three space dimensions exactly. We also introduce a second-order, unsplit, volume-of-fluid advection algorithm that is based on a second-order, finite difference method for scalar conservation laws due to Bell, Dawson and Shubin. We test this advection algorithm by modeling several different interface shapes propagating in two simple incompressible flows and compare the results with the standard second-order, operator-split advection algorithm. Although both methods are second-order accurate when the interface is smooth, we find that the unsplit algorithm exhibits noticeably better resolution in regions where the interface has discontinuous derivatives, such as at corners.

This paper extends the treatment of discontinuities introduced by the author in [13], [14], and [15] to the two dimensional case. The main idea relies on the fact that on each side of a discontinuity the computations draw information from the same side. A numerical method for ordinary differential equations modeling the movement of the discontinuity curve is incorporated into the algorithm to compute discontinuity positions. The conservation feature of the treatment is studied for the case of an isolated discontinuity. Finally, we study two-dimensional scalar systems and systems of conservation laws and display some numerical results when our treatment is applied.

In this paper we consider scalar hyperbolic equations in one space dimension of the type ut(x,t)+ddxf(u;x)=h(u;x) , u(x,0)=u0(x), x∈Rt>0 , where ƒ ϵ C1 and h continuous w.r.t. u, x. The initial condition is assumed to be piecewise continuous. We present a new method for constructing the entropy solution of (1) at a fixed time t = T > 0 in one time step based on transporting the initial values along characteristics. If the solution of (1) is smooth, we get the exact solution; in case of shocks the multivalued graph of the initial data is corrected by a geometrical averaging technique via the conservation principle. The method is also applicable to a scalar equation in which there is a mild coupling between the physical dimensions in the problem, for example, ut(x,y)+ddxf(u;x,y)+ddyf(u;x,y)=h(u;x,y) . By a change of variables, (2) can be reduced to a quasi one-dimensional problem. We conjecture that the advantage of computing the entropy solution at a fixed time in one time step cannot easily be carried over to systems. But we have some hints that this might be possible in case of scalar equations in two space dimensions with arbitrary fluxes ƒ1, ƒ2. The CPU time depends only on the total number of shocks which occur in the entropy solution up to time T; the accuracy of the computed shock position is of order at least 10−2. Since our method is not based on a time discretisation, questions (and problems) concerning stability and convergence do not arise.