Semi-discrete Form Research Articles

The soliton solutions for semidiscrete complex mKdV equation

The semidiscrete complex modified Korteweg–de Vries equation (semidiscrete cmKdV), which is the second member of the semidiscrete nonlinear Schrődinger hierarchy (Ablowitz–Ladik hierarchy), is solved using the Hirota bilinear formalism. Starting from the focusing case of semidiscrete form of cmKdV, proposed by Ablowitz and Ladik, we construct the bilinear form and build the multi-soliton solutions. The complete integrability of semidiscrete cmKdV, focusing case, is proven and results are discussed.

Blow-up for Discretizations of a Nonlinear Parabolic Equation With Nonlinear Memory and Mixt Boundary Condition

In this paper, we study the numerical approximation for the following initial-boundary value problem
 
 v_t=v_{xx}+v^q\int_{0}^{t}v^p(x,s)ds, x\in(0,1), t\in(0,T)
 
 v(0,t)=0, v_x(1,t)=0, t\in(0,T)
 
 v(x,0)=v_0(x)>0}, x\in(0,1)
 
 where q>1, p>0. Under some assumptions, it is  shown that the solution of a semi-discrete form of this problem blows up in the finite time and estimate its semi-discrete blow-up time. We also prove that the semi-discrete blows-up time converges to the real one when the mesh size goes to zero. A similar study has been also undertaken for a discrete form of the above problem. Finally, we give some numerical results to illustrate our analysis.

Multiscale Polynomial-Based High-Order Central High Resolution Schemes

Polynomial-based high order central high resolution schemes with semi-discrete forms are integrated with multiresolution-based adapted cells/grids. To preserve the positivity condition on non-uniform cells/grids, the corresponding formulations are studied, redesigned or developed. Two general approaches can be used for polynomial-based reconstructions: (a) direct interpolation by a polynomial, (b) proper combination of different polynomials to construct a new polynomial with desired features. Based on these approaches, three polynomial-based reconstructions are considered: (i) parabolic polynomials interpolating average solutions of three successive cells; (ii) piece-wise parabolic methods (PPMs) obtained with two different local stencils; (iii) central-WENO schemes [based on the results of approach (i)]. For the first approach, the corresponding features, stability conditions, formulations and nonlinear limiters are studied and updated. For the second approach, for more localized stencils, new independent variables (e.g., first and second spatial derivatives) are introduced by adding new conservation laws. Two PPM-based central schemes are formulated and a new limiter and a new updating procedure are introduced. For the third approach, the average-interpolating parabolic polynomial [in approach (i)] is used in the framework of the central-WENO formulation. Third and fourth order formulations are provided on non-uniform grids/cells. Finally some numerical examples are presented to verify the results and to assess effectiveness and robustness of the three approaches.

Energy conserving Galerkin approximation of two dimensional wave equations with random coefficients

Wave propagation problems for heterogeneous media are known to have many applications in physics and engineering. Recently, there has been an increasing interest in stochastic effects due to the uncertainty, which may arise from impurities of the media. This work considers a two-dimensional wave equation with random coefficients which may be discontinuous in space. Generalized polynomial chaos method is used in conjunction with stochastic Galerkin approximation, and local discontinuous Galerkin method is used for spatial discretization. Our method is shown to be energy preserving in semi-discrete form as well as in fully discrete form, when leap-frog time discretization is used. Its convergence rate is proved to be optimal and the error grows linearly in time. The theoretical properties of the proposed scheme are validated by numerical tests.

A semi-implicit integration factor discontinuous Galerkin method for the non-linear heat equation

In this paper, a new discontinuous Galerkin method is employed to study the non-linear heat conduction equation with temperature dependent thermal conductivity. We present practical implementation of the new discontinuous Galerkin scheme with weighted flux averages. The second-order implicit integration factor for time discretization method is applied to the semi discrete form. We obtain the L2 stability of the discontinuous Galerkin scheme. Numerical examples show that the error estimates are of second order when linear element approximations are applied. The method is applied to the non-linear heat conduction equations with source term.

Numerical Quenching for Heat Equations with Coupled Nonlinear Boundary Flux

In this paper, we study a numerical approximation of the following problem ut = uxx, vt = vxx, 0 < x < 1, 0 < t < T; ux(0, t) = u−m(0, t) + v−p(0, t), vx(0, t) = u−q (0, t) + v−n(0, t) and ux(1, t) = vx(1, t) = 0, 0 < t < T, where m, p, q and n are parameters. We prove that the solution of a semidiscrete form of above problem quenches in a finite time only at first node of the mesh. We show that the time derivative of the solution blows up at quenching node. Some conditions under which the non-simultaneous or simultaneous quenching occurs for the solution of the semidiscrete problem are obtained. We establish the convergence of the quenching time. Finally, some numerical results to illustrate our analysis are given.

Diffuse interface models for two-phase flows in artificial compressibility approach

We present an approach to numerical simulation of two-phase flows in the so-called one-fluid formulation, combining the Entropically Damped Artificial Compressibility (EDAC) method for flow solution and a chosen variant of the Phase Field Method (PFM) that belongs to a wider family of the Diffuse Interface (DI) methods of interface capturing. The resulting governing equations express a set of conservation laws and are of the advection–diffusion type, convenient in numerical handling. These equations, rewritten in the semi-discrete form, can be efficiently solved on parallel devices using the method of lines. We applied the conservative finite difference method for the spatial discretisation and a variant of the Runge–Kutta method to advance the solution in time. The results presented in this work contain the analysis of spurious currents in the case of stationary droplet together with the pressure jump across its surface, as well as the deformation of a droplet subjected to the laminar Couette flow. The presented EDAC–DI approach, offering very high computational efficiency, gives results comparable to other well-established methods for the simulations of the interfacial flows.

A Hybrid Staggered Discontinuous Galerkin Method for KdV Equations

A hybrid staggered discontinuous Galerkin method is developed for the Korteweg–de Vries equation. The equation is written into a system of first order equations by introducing auxiliary variables. Two sets of finite element functions are introduced to approximate the solution and the auxiliary variables. The staggered continuity of the two finite element function spaces gives a natural flux condition and trace value on the element boundaries in the derivation of Galerkin approximation. On the other hand, to deal with the third order derivative term an hybridization idea is used and additional flux unknowns are introduced. The auxiliary variables can be eliminated in each element and the resulting algebraic system on the solution and the additional flux unknowns is solved. Stability of the semi discrete form is proven for various boundary conditions. Numerical results present the optimal order of $$L^2$$ -errors of the proposed method for a given polynomial order.

A new high-order scheme based on numerical dispersion analysis of the wave phase velocity for semidiscrete wave equations

In the numerical computation of wave equations, numerical dispersion is a persistent problem arising from inadequate discretization of the continuous wave equation. To thoroughly understand the mechanism of numerical dispersion, we separately analyze the numerical dispersion relations of time-stepping and spatial discretization schemes by Fourier analysis. The relevant results show that the numerical dispersion errors of time-marching schemes depend on the time step length or the Courant number, whereas the numerical dispersion errors of spatial discretization schemes are determined by the error between the eigenvalues of the numerical spatial differential operator and the continuous spatial differential operator. We also find that the much better numerical dispersion accuracy of the stereo-modeling discrete (SMD)-type operator can be attributed to the inclusion of diversified basis functions for its eigenvalue. Based on these findings, we combine the optimal four-stage symplectic partitioned Runge-Kutta and eighth-order SMD as a new fully discrete scheme. The subsequent analysis of its normalized phase velocity is consistent with the numerical dispersion analysis in semidiscrete forms. This is followed by an acoustic wave simulation in a homogeneous model and corresponding computational efficiency comparison. The results show that the new scheme is much more accurate and antidispersive on a coarse grid. In the final two numerical experiments, we use the new scheme to model the acoustic wave propagation in a three-layer model and the Marmousi model. The convolutional perfectly matched layer is applied to eliminate artificial boundary reflections. Our semidiscrete numerical dispersion analysis provides an efficient tool to quantitatively evaluate the time-stepping and spatial discretization schemes. It can facilitate the development of more accurate fully discrete numerical schemes for solving seismic wave equations.

Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems

The objective of this paper is to present a local bounds preserving stabilized finite element scheme for hyperbolic systems on unstructured meshes based on continuous Galerkin (CG) discretization in space. A CG semi-discrete scheme with low order artificial dissipation that satisfies the local extremum diminishing (LED) condition for systems is used to discretize a system of conservation equations in space. The low order artificial diffusion is based on approximate Riemann solvers for hyperbolic conservation laws. In this case we consider both Rusanov and Roe artificial diffusion operators. In the Rusanov case, two designs are considered, a nodal based diffusion operator and a local projection stabilization operator. The result is a discretization that is LED and has first order convergence behavior. To achieve high resolution, limited antidiffusion is added back to the semi-discrete form where the limiter is constructed from a linearity preserving local projection stabilization operator. The procedure follows the algebraic flux correction procedure usually used in flux corrected transport algorithms. To further deal with phase errors (or terracing) common in FCT type methods, high order background dissipation is added to the antidiffusive correction. The resulting stabilized semi-discrete scheme can be discretized in time using a wide variety of time integrators. Numerical examples involving nonlinear scalar Burgers equation, and several shock hydrodynamics simulations for the Euler system are considered to demonstrate the performance of the method. For time discretization, Crank–Nicolson scheme and backward Euler scheme are utilized.

On an efficient second order backward difference Newton scheme for MHD system

For the 2D/3D time-dependent magnetohydrodynamics (MHD) system, we propose a new second order backward difference formula Newton scheme (SBDFN) which is a combination of second order backward difference approximation of the time derivative terms and Newton treatment of the nonlinear terms. Meanwhile, the finite element method is applied as spatial discretization. Firstly, the optimal convergence of spatially semi-discrete form is deduced. Secondly, the stability and well-posedness of SBDFN scheme are provided under τ<C, where C is independent of h. Based on these results, optimal error estimates of the scheme in time are proved by negative norm technique. Finally, numerical experiments are carried out to validate the theoretical analysis.

Galerkin finite element method for two-dimensional space and time fractional Bloch–Torrey equation

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 L2−1σ formula to discretize temporal Caputo derivative, we obtain the local truncation error with O(τ3−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 L2 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.

Analysis and approximation of a fractional Laplacian-based closure model for turbulent flows and its connection to Richardson pair dispersion

We study a turbulence closure model in which the fractional Laplacian (−Δ)α of the velocity field represents the turbulence diffusivity. We investigate the energy spectrum of the model by applying Pao’s energy transfer theory. For the case α=1∕3, the corresponding power law of the energy spectrum in the inertial range has a correction exponent on the regular Kolmogorov −5∕3 scaling exponent. For this case, this model represents Richardson’s particle pair-distance superdiffusion of a fully developed homogeneous turbulent flow as well as Lévy jumps that lead to the superdiffusion. For other values of α, the power law of the energy spectrum is consistent with the regular Kolmogorov −5∕3 scaling exponent. We also propose and study a modular time-stepping algorithm in semi-discretized form. The algorithm is minimally intrusive to a given legacy code for solving Navier–Stokes equations by decoupling the local part and nonlocal part of the equations for the unknowns. We prove the algorithm is first-order accurate and unconditionally stable. We also derive error estimates for full discretizations of the model which, in addition to the time stepping algorithm, involves a finite element spatial discretization and a domain truncation approximation to the range of the fractional Laplacian.

A central-upwind geometry-preserving method for hyperbolic conservation laws on the sphere

We introduce a second-order, central-upwind finite volume method for the discretization of nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. The semi-discrete version of the proposed method is based on a technique of local propagation speeds and it is free of any Riemann solver. The main advantages of our scheme are the high resolution of discontinuous solutions, its low numerical dissipation, and its simplicity for the implementation. The proposed scheme does not use any splitting approach, which is applied in some cases to upwind schemes in order to simplify the resolution of Riemann problems. The semi-discrete form of the scheme is strongly linked to the analytical properties of the nonlinear conservation law and to the geometry of the sphere. The curved geometry is treated here in an analytical way so that the semi-discrete form of the proposed scheme is consistent with a geometric compatibility property. Furthermore, the time evolution is carried out by using a total-variation-diminishing Runge-Kutta method. A rich family of (discontinuous) stationary solutions is available for the problem under consideration when the flux is nonlinear and foliated (as identified by the author in an earlier work). We present here a series of numerical examples, obtained by considering non-trivial steady state solutions and this leads us to a good validation of the accuracy and efficiency of the proposed central-upwind finite volume method. Our numerical tests confirm the stability of the proposed scheme and clearly show its ability to capture accurately discontinuous steady state solutions to nonlinear hyperbolic conservation laws posed on the sphere.

Superconvergence analysis of Yee scheme for metamaterial Maxwell’s equations on non-uniform rectangular meshes

Since the development of Yee scheme back in 1966, it has become one of the most popular simulation tools for modeling electromagnetic wave propagation in various situations. However, its rigorous error analysis on nonuniform rectangular type grids was carried out until 1994 by Monk and Suli. They showed that the Yee scheme is still second-order convergent on a nonuniform mesh even though the local truncation error is only of first order. In this paper, we extend their results to Maxwell's equations in metamaterials by a simpler proof, and show the second-order superconvergence in space for the true Yee scheme instead of the only semi-discrete form discussed in Monk and Suli's original work. Numerical results supporting our analysis are presented.

Anchored analysis of variance Petrov–Galerkin projection schemes for linear stochastic structural dynamics

In this paper, we propose anchored functional analysis of variance Petrov–Galerkin (AAPG) projection schemes, originally developed in the context of parabolic stochastic partial differential equations (Audouze C, Nair PB. 2014 Comput. Methods Appl. Mech. Eng. 276, 362–395. ( doi:10.1016/j.cma.2014.02.023 )) for solving a class of problems in linear stochastic structural dynamics. We consider the semi-discrete form of the governing equations in the time-domain and the proposed formulation involves approximating the dynamic response using a Hoeffding functional analysis of variance decomposition. Subsequently, we design a set of test functions for a stochastic Petrov–Galerkin projection scheme that enables the original high-dimensional problem to be decomposed into a sequence of decoupled low-dimensional subproblems that can be solved independently of each other. Numerical results are presented to demonstrate the efficiency and accuracy of AAPG projection schemes and comparisons are made to approximations obtained using Monte Carlo simulation, generalized polynomial chaos-based stochastic Galerkin projection schemes and the generalized spectral decomposition method. The results obtained suggest that the proposed approach holds significant potential for alleviating the curse of dimensionality encountered in tackling high-dimensional problems in stochastic structural dynamics with a large number of spatial and stochastic degrees of freedom.

Weakly corrected numerical solutions to stochastically driven nonlinear dynamical systems

A method to weakly correct the solutions of stochastically driven nonlinear dynamical systems, herein numerically approximated through the Eule–Maruyama (EM) time-marching map, is proposed. An essential feature of the method is a change of measures that aims at rendering the EM-approximated solution measurable with respect to the filtration generated by an appropriately defined error process. Using Ito's formula and adopting a Monte Carlo (MC) setup, it is shown that the correction term may be additively applied to the realizations of the numerically integrated trajectories. Numerical evidence, presently gathered via applications of the proposed method to a few nonlinear mechanical oscillators and a semi-discrete form of a 1-D Burger's equation, lends credence to the remarkably improved numerical accuracy of the corrected solutions even with relatively large time step sizes.

Spectral Deferred Correction Method for FDTD

Finite difference time domain (FDTD) is widely used to compute the transient behavior of transmission lines. However, the time step of classic FDTD is limited by the Courant-Friedrich-Lewy (CFL) condition, which makes it not sufficiently efficient for relative low frequency (compared with computation cell size and wave length) signals. The FDTD with alternating direction implicit (ADI) technique is not restricted by the CFL condition, but suffers from numerical dispersions. This paper employs the spectral deferred correction method as the time marching scheme to solve the semidiscretized form of the telegraph equation. The method is unconditionally stable, which allows a larger time step, and can be essentially arbitrary high order accurate.

Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations

In this paper, a class of two-dimensional space and time fractional Bloch–Torrey equations (2D-STFBTEs) are considered. Some definitions and properties of fractional derivative spaces are presented. By finite difference method and Galerkin finite element method, a semi-discrete variational formulation for 2D-STFBTEs is obtained. The stability and convergence of the semi-discrete form are discussed. Then, a fully discrete scheme of 2D-STFBTEs is derived and the convergence is investigated. Finally, some numerical examples based on linear piecewise polynomials and quadratic piecewise polynomials are given to prove the correctness of our theoretical analysis.

Singular integrals, scale-space and wavelet transforms

The Gaussian scale-space is a singular integral convolution operator with scaled Gaussian kernel. For a large class of singular integral convolution operators with differentiable kernels, a general method for constructing mother wavelets for continuous wavelet transforms is developed, and Calderón type inversion formulas, in both integral and semi-discrete forms, are derived for functions in Lp spaces. In the case of the Gaussian scale-space, the semi-discrete inversion formula can further be expressed as a sum of wavelet transforms with the even order derivatives of the Gaussian as mother wavelets. Similar results are obtained for B-spline scale-space, in which the high frequency component of a function between two consecutive dyadic scales can be represented as a finite linear combination of wavelet transforms with the derivatives of the B-spline or the spline framelets of Ron and Shen as mother wavelets.