In this paper, we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations. The finite difference method is employed to approximate the multi-dimensional Riesz fractional derivatives, which generates symmetric positive definite ill-conditioned multi-level Toeplitz matrices. The preconditioned conjugate gradient method with a preconditioner based on the sine transform is employed to solve the resulting linear system. Theoretically, we prove that the spectra of the preconditioned matrices are uniformly bounded in the open interval $(\frac{1}{2},\frac{3}{2})$ and thus the preconditioned conjugate gradient method converges linearly within an iteration number independent of the discretization step-size. Moreover, the proposed method can be extended to handle ill-conditioned multi-level Toeplitz matrices whose blocks are generated by functions with zeros of fractional order. Our theoretical results fill in a vacancy in the literature. Numerical examples are presented to show the convergence performance of the proposed preconditioner that is better than other preconditioners.

Linear complementarity problems have drawn considerable attention in recent years due to their wide applications. In this article, we introduce the two-step two-sweep modulus-based matrix splitting (TSTM) iteration method and two-sweep modulus-based matrix splitting type II (TM II) iteration method which are a combination of the two-step modulus-based method and the two-sweep modulus-based method, as two more effective ways to solve the linear complementarity problems. The convergence behavior of these methods is discussed when the system matrix is either a positive-definite or an $H_+$-matrix. Finally, numerical experiments are given to show the efficiency of our proposed methods.

In this paper, we investigate the long-time near-conservations of energy and kinetic energy by the widely used exponential integrators to highly oscillatory conservative systems. The modulated Fourier expansions of two kinds of exponential integrators have been constructed and the long-time numerical conservations of energy and kinetic energy are obtained by deriving two almost-invariants of the expansions. Practical examples of the methods are given and the theoretical results are confirmed and demonstrated by a numerical experiment.

This paper studies a problem of optimal investment and consumption with early retirement option under constant elasticity variation (CEV) model with finite horizon. Two risky assets are involved in the model with one following geometric Brownian motion and the other a CEV model. This problem is a kind of two dimensional mixed control and optimal stopping problems with finite horizon. The existence and continuity of the optimal retirement threshold surfaces are proved and the working and retirement regions are characterized theoretically. Least-squares Monte-Carlo methods are developed to solve this mixed control and optimal stopping problem. The algorithms are well implemented and the optimal retirement threshold surfaces, optimal investment strategies and the optimal consumptions are drawn via examples.

This paper develops a posteriori error bound for a space-time finite element method for the linear wave equation. The standard $P_l$ conforming element is used for the spatial discretization and a $P_2$-CDG method is applied for the time discretization. The essential ingredients in the a posteriori error analysis are the twice time reconstruction functions and the $C^1(J)$-smooth elliptic reconstruction, which lead to reliable a posteriori error bound in view of the energy method. As an outcome, a time adaptive algorithm is proposed with the error equidistribution strategy. Numerical experiments are reported to illustrate the performance of the a posteriori error bound and the validity of the adaptive algorithm.

Time-dependent fractional partial differential equations typically require huge amounts of memory and computational time, especially for long-time integration, which taxes computational resources heavily for high-dimensional problems. Here, we first analyze existing numerical methods of sum-of-exponentials for approximating the kernel function in constant-order fractional operators, and identify the current pitfalls of such methods. In order to overcome the pitfalls, an improved sum-of-exponentials is developed and verified. We also present several sum-of-exponentials for the approximation of the kernel function in variable-order fractional operators. Subsequently, based on the sum-of-exponentials, we propose a unified framework for fast time-stepping methods for fractional integral and derivative operators of constant and variable orders. We test the fast method based on several benchmark problems, including fractional initial value problems, the time-fractional Allen-Cahn equation in two and three spatial dimensions, and the Schrödinger equation with nonreflecting boundary conditions, demonstrating the efficiency and robustness of the proposed method. The results show that the present fast method significantly reduces the storage and computational cost especially for long-time integration problems.

We propose a hybrid scheme combing the level set method and the multi-component diffuse interface method to simulate complex multi-phase flows. The overall numerical scheme is based on a sharp interface framework where the level set method is adopted to capture the material interface, the Euler equation is used to describe a single-phase flow on one side of the interface and the six-equation diffuse interface model is applied to model the multi-phase mixture or gas-liquid cavitation on the other side. An exact Riemann solver, between the Euler equation and the six-equation model with highly nonlinear Mie-Grüneisen equations of state, is developed to predict the interfacial states and compute the phase interface flux. Several numerical examples, including shock tube problems, cavitation problems, air blast and underwater explosion applications are presented to validate the numerical scheme and the Riemann solver.

We make the split of the integral fractional Laplacian as $$(−∆)^su = (−∆)(−∆)^{s−1}u,$$ where $s ∈ (0,\frac{1}{2}) ∪ (\frac{1}{2}, 1).$ Based on this splitting, we respectively discretize the one- and two-dimensional integral fractional Laplacian with the inhomogeneous Dirichlet boundary condition and give the corresponding truncation errors with the help of the interpolation estimate. Moreover, the suitable corrections are proposed to guarantee the convergence in solving the inhomogeneous fractional Dirichlet problem and an $\mathcal{O}(h ^{1+α−2s})$ convergence rate is obtained when the solution $u ∈ C ^{1,α}(\overline{Ω}^ δ_n),$ where $n$ is the dimension of the space, $α ∈ ({\rm max}(0, 2s − 1), 1], δ$ is a fixed positive constant, and $h$ denotes mesh size. Finally, the performed numerical experiments confirm the theoretical results.

In this paper, we present a quadratic auxiliary variable approach to develop a new class of energy-preserving Runge-Kutta methods for the Korteweg-de Vries equation. The quadratic auxiliary variable approach is first proposed to reformulate the original model into an equivalent system, which transforms the energy conservation law of the Korteweg-de Vries equation into two quadratic invariants of the reformulated system. Then the symplectic Runge-Kutta methods are directly employed for the reformulated model to arrive at a new kind of time semi-discrete schemes for the original problem. Under consistent initial conditions, the proposed methods are rigorously proved to maintain the original energy conservation law of the Korteweg-de Vries equation. In addition, the Fourier pseudo-spectral method is used for spatial discretization, resulting in fully discrete energy-preserving schemes. To implement the proposed methods effectively, we present a very efficient iterative technique, which not only greatly saves the calculation cost, but also achieves the purpose of practically preserving structure. Ample numerical results are addressed to confirm the expected order of accuracy, conservative property and efficiency of the proposed algorithms.

Orthogonal matching pursuit (OMP for short) algorithm is a popular method of sparse signal recovery in compressed sensing. This paper applies OMP to the sparse polynomial reconstruction problem. Distinguishing from classical research methods using mutual coherence or restricted isometry property of the measurement matrix, the recovery guarantee and the success probability of OMP are obtained directly by the greedy selection ratio and the probability theory. The results show that the failure probability of OMP given in this paper is exponential small with respect to the number of sampling points. In addition, the recovery guarantee of OMP obtained through classical methods is lager than that of $ℓ_1$-minimization whatever the sparsity of sparse polynomials is, while the recovery guarantee given in this paper is roughly the same as that of $ℓ_1$-minimization when the sparsity is less than 93. Finally, the numerical experiments verify the availability of the theoretical results.

Compact higher-order (HO) schemes for a new finite difference method, referred to as the Cartesian cut-stencil FD method, for the numerical solution of the convection-diffusion equation in complex shaped domains have been addressed in this paper. The Cartesian cut-stencil FD method, which employs 1-D quadratic transformation functions to map a non-uniform (uncut or cut) physical stencil to a uniform computational stencil, can be combined with compact HO Padé-Hermitian formulations to produce HO cut-stencil schemes. The modified partial differential equation technique is used to develop formulas for the local truncation error for the cut-stencil HO formulations. The effect of various HO approximations for Neumann boundary conditions on the solution accuracy and global order of convergence are discussed. The numerical results for second-order and compact HO formulations of the Cartesian cut-stencil FD method have been compared for test problems using the method of manufactured solutions.