A linear system from finite difference discretization of a generalized nonlocal elastic model was studied, where the model is composed of a Riesz potential operator with a fractional differential operator. Some properties of the coefficient matrix are proven theoretically and it is found that the linear system is very ill-conditioned when the parameter in the long-range hydrodynamic interactions is close to zero. Therefore, the usual Krylov subspace method with the Strang-Strang circulant preconditioner loses the power of preconditioning so that the iterative method converges slowly. Here the problem is fixed by utilizing a mixed-type circulant preconditioner obtained by both Strang’s and Chan’s circulant approximations. The invertibility of the preconditioner and a small-norm-low-rank decomposition of the difference matrix of the coefficient matrix and the preconditioner are shown theoretically under certain conditions. Numerical examples are given to illustrate the efficiency of the proposed fast solver.

A time-fractional diffusion equation with an interface problem caused by discontinuous coefficients is considered. To solve it, in the temporal direction Alikhanov’s $L2-1_σ$ method with graded mesh is presented to deal with the weak singularity at $t = 0,$ while in the spatial direction a finite element method with uniform mesh is employed to handle the discontinuous coefficients. Then, with the help of discrete fractional Grönwall inequality and the robustness theory of $α → 1^−,$ we show that the method has stable error bounds at $α → 1^−,$ the fully discrete schemes $L^2(Ω)$ norm and $H^1(Ω)$ semi-norm are unconditionally stable, and the optimal convergence order is $\mathscr{O}(h^2 + N^{−{\rm min}\{rα,2\}})$ and $\mathscr{O}(h + N^{−{\rm min}\{rα,2\}}),$ respectively, where, $h,$ $N,$ $α,$ $r$ is the total number of spatial parameter, the time-fractional order coefficient, and the time grid constant. Finally, three numerical examples are provided to illustrate our theoretical results.

We qualitatively analyze the bounded traveling wave solutions of a (2+1)- dimensional generalized breaking soliton (gBS) equation by using the theory of planar dynamical systems. We present the global phase diagrams of the dynamical system corresponding to the (2+1)-dimensional gBS equation under different parameters. The conditions for the existence of bounded traveling wave solutions are successfully derived. We find the relationship between the waveform of bounded traveling wave solutions and the dissipation coefficient $β.$ When the absolute value of the dissipation coefficient $β$ is greater than a critical value, we find that the equation has a kink profile solitary wave solution, while the solution has oscillatory and damped property if $|β|$ is less than the critical value. In addition, we give the exact bell profile solitary wave solution and kink profile solitary wave solution by using undetermined coefficient method. The approximate oscillatory damped solution is given constructively. Through error analysis, we find that the approximate oscillatory damped solution is meaningful. Finally, we present the graphical analysis of the influence of dissipation coefficient $β$ on oscillatory damped solution in order to better understand their dynamical behaviors.

An American exchange option is a rainbow option with two underlying assets, whose pricing model is a two-dimensional free boundary problem and is equivalent to a parabolic variational inequality problem on a two-dimensional unbounded domain. The present work proposes an effective numerical method for this complex problem. We first reduce the problem into a one-dimensional linear complementarity problem (LCP) on a bounded domain based on a dimension reduction transformation, an a priori estimate for the optimal exercise boundary, and a far-field truncation technique. This LCP is then approximated by a finite element method with a geometric partition in the spatial direction and a backward Euler method with a uniform partition in the temporal direction. The convergence order of the fully discretized scheme is established as well. Further, according to the features of the discretized system, a primal-dual active-set (PDAS) method is imposed to solve this problem to obtain the option price and the optimal exercise boundary simultaneously. Finally, several numerical simulations are carried out to verify the theoretical results and effectiveness of the proposed method.

Unconditionally stable and convergent variable-step BDF2 scheme for solving the MBE model with slope selection is derived. Discrete orthogonal convolution kernels of the variable-step BDF2 method are commonly utilized for solving the phase field models. We present new inequalities, concerning the vector forms, for the kernels especially dealing with nonlinear terms in the slope selection model. The convergence rate of the fully discrete scheme is proved to be two both in time and space in $L^2$ norm under the setting of the variable time steps. Energy dissipation law is proved rigorously with a modified energy by adding a small term to the discrete version of the original free energy functional. Two numerical examples including an adaptive time-stepping strategy are given to verify the convergence rate and the energy dissipation law.

Recently, a fast deterministic block Kaczmarz (FDBK) method which uses a greedy criterion of the row selections and contains pseudoinverse-free computation is presented. In this work, we introduce a maximum residual rule into FDBK and develop a new block Kaczmarz method which is also considered as a fast deterministic pseudoinverse-free block extension of Motzkin (FBEM) method. In addition, we prove that FBEM converges linearly to the unique least-norm solution of the linear systems. Furthermore, by incorporating the Polyak momentum technique into the FBEM iteration method, we establish an accelerated variant of FBEM (mFBEM) and show its global linear convergence. Numerical examples using artificial and real datasets demonstrate the effectiveness of FBEM as well as mFBEM.

Arbitrary high-order numerical schemes conserving the momentum and energy of the generalized Rosenau-type equation are studied. Derivation of momentum-preserving schemes is made within the symplectic Runge-Kutta method coupled with the standard Fourier pseudo-spectral method in space. Combining quadratic auxiliary variable approach, symplectic Runge-Kutta method, and standard Fourier pseudo-spectral method, we introduce a class of high-order mass- and energy-preserving schemes for the Rosenau equation. Various numerical tests illustrate the performance of the proposed schemes.

We consider polynomials orthogonal with respect to a nonstandard inner product. In fact, we deal with Sobolev-type orthogonal polynomials in the broad sense of the expression. This means that the inner product under consideration involves the Hahn difference operator, thus including the difference operators $\mathscr{D}_q$ and $∆$ and, as a limit case, the derivative operator. In a previous work, we studied properties of these polynomials from a theoretical point of view. There, we obtained a second-order differential/difference equation satisfied by these polynomials. The aim of this paper is to present an algorithm and a symbolic computer program that provides us with the coefficients of the second-order differential/difference equation in this general context. To illustrate both, the algorithm and the program, we will show three examples related to different operators.

We establish a uniform error bound of an exponential wave integrator Fourier pseudospectral (EWI-FP) method for the long-time dynamics of the nonlinear Schrödinger equation with wave operator (NLSW), in which the strength of the nonlinearity is characterized by $\varepsilon^{2p}$ with $\varepsilon\in(0,1]$ a dimensionless parameter and $p\in \mathbb{N}^+$. When $0<\varepsilon\ll 1,$ the long-time dynamics of the problem is equivalent to that of the NLSW with $\mathscr{O}(1)$-nonlinearity and $\mathscr{O}(\varepsilon)$-initial data. The NLSW is numerically solved by the EWI-FP method which combines an exponential wave integrator for temporal discretization with the Fourier pseudospectral method in space. We rigorously establish the uniform $H^1$-error bound of the EWI-FP method at $\mathscr{O}(h^{m-1}+\varepsilon^{2p-\beta}\tau^2)$ up to the time at $\mathscr{O}(1/\varepsilon^\beta)$ with $0\le \beta \le 2p$, the mesh size $h,$ time step $\tau$ and $m ≥ 2$ an integer depending on the regularity of the exact solution. Finally, numerical results are provided to confirm our error estimates of the EWI-FP method and show that the convergence rate is sharp.