We propose a numerical method to solve the Monge-Ampère equation which admits a classical convex solution. The Monge-Ampere equation is reformulated into an equivalent first-order system. We adopt a novel reconstructed discontinuous approximation space which consists of piecewise irrotational polynomials. This space allows us to solve the first-order system in two sequential steps. In the first step, we solve a nonlinear system to obtain the approximation to the gradient. A Newton iteration is adopted to handle the nonlinearity of the system. The approximation to the primitive variable is obtained from the approximate gradient by a trivial least squares finite element method in the second step. Numerical examples in both two and three dimensions are presented to show an optimal convergence rate in accuracy. It is interesting to observe that the approximation solution is piecewise convex. Particularly, with the reconstructed approximation space, the proposed method numerically demonstrates a remarkable robustness. The convergence of the Newton iteration does not rely on the initial values. The dependence of the convergence on the penalty parameter in the discretization is also negligible, in comparison to the classical discontinuous approximation space.

In this paper, we propose a variational multiscale method (VMM) for the stationary incompressible magnetohydrodynamics equations. This method is defined by large-scale spaces for the velocity field and the magnetic field, which aims to solve flows at high Reynolds numbers. We provide a new VMM formulation and prove its stability and convergence. Finally, some numerical experiments are presented to indicate the optimal convergence of our method.

The Richtmyer–Meshkov instability of interfaces separating elastic-plastic materials from vacuum is investigated by numerical simulation using a multi-material solid mechanics algorithm based on an Eulerian framework. The research efforts are directed to reveal the influence of the initial perturbation and material strength on the deformation of the perturbed interface impacted by an initial shock. By varying the initial amplitude $(kξ_0)$ of the perturbed interface and the yield stress $(σ_{\Upsilon}),$ three typical modes of interface deformation have been identified as the broken mode, the stable mode and the oscillating mode. For the broken mode, the interface width (i.e., the bubble position with respect to that of the spike) increases continuously resulting in a final separation of the spike from the perturbed interface. For the stable mode, the interface width grows to saturation and then maintains a nearly constant value in the long term. For the oscillating mode, the wavy-like interface moving forward obtains an aperiodic oscillation of small amplitude, namely, the interface width varies in time slightly around zero. The intriguing difference of the typical modes is interpreted qualitatively by comparing the early-stage wave motion and the commensurate pressure and effective stress. Further, the subsequent interface deformation is illustrated quantitatively via the time series of the interface positions and velocities of these three typical modes.

The background numerical noise $ε_0$ is determined by the maximum of truncation error and round-off error. For a chaotic system, the numerical error $ε(t)$ grows exponentially, say, $ε(t) = ε_0 {\rm exp}(κt),$ where $κ > 0$ is the so-called noise-growing exponent. This is the reason why one can not gain a convergent simulation of chaotic systems in a long enough interval of time by means of traditional algorithms in double precision, since the background numerical noise $ε_0$ might stop decreasing because of the use of double precision. This restriction can be overcome by means of the clean numerical simulation (CNS), which can decrease the background numerical noise $ε_0$ to any required tiny level. A lot of successful applications show the novelty and validity of the CNS. In this paper, we further propose some strategies to greatly increase the computational efficiency of the CNS algorithms for chaotic dynamical systems. It is highly suggested to keep a balance between truncation error and round-off error and besides to progressively enlarge the background numerical noise $ε_0,$ since the exponentially increasing numerical noise $ε(t)$ is much larger than it. Some examples are given to illustrate the validity of our strategies for the CNS.

This investigation focused on the influence of the radial inertia effect on the propagation behavior of stress waves in thin-walled tubes subjected to combined longitudinal and torsional impact loads. Generalized characteristics theory was used to analyze the main features of the characteristic wave speeds and simple wave solutions in thin-walled tubes. The incremental elastic-plastic constitutive relations described by the rate-independent plasticity were adopted, and the finite difference method was used to investigate the evolution and propagation behaviors of combined elastic-plastic stress waves in thin-walled tubes when the radial inertial effect was considered. The numerical results were compared with those obtained when the radial inertia effect was not considered. The results showed that the speed of the coupled stress wave increased when the radial inertia effect was considered. The hardening modulus of the material in the plastic stage had a greater impact on the coupled slow waves than on the coupled fast waves.

In this paper, we present a quadratic auxiliary variable (QAV) technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation. The QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial condition. Then the reformulated QAV system is discretized by applying the Fourier pseudo-spectral method in space and the symplectic Runge-Kutta methods in time, which arrives at a class of fully discrete schemes. Under the consistent initial condition, they can be rewritten as a new fully discrete system by eliminating the introduced auxiliary variable, which is rigorously proved to be energy-preserving and symmetric. Ample numerical experiments are conducted to confirm the expected order of accuracy, conservative property and efficiency of the proposed methods. The presented numerical strategy makes it possible to directly apply a special class of Runge-Kutta methods to develop energy-preserving algorithms for a general conservative system with any polynomial energy.

In this article, a robust, effective, and scale-invariant weighted compact nonlinear scheme (WCNS) is proposed by introducing descaling techniques to the nonlinear weights of the WCNS-Z/D schemes. The new scheme achieves an essentially non-oscillatory approximation of a discontinuous function (ENO-property), a scale-invariant property with an arbitrary scale of a function (Si-property), and an optimal order of accuracy with smooth function regardless of the critical point (Cp-property). The classical WCNS-Z/D schemes do not satisfy Si-property intrinsically, which is caused by a loss of sub-stencils’ adaptivity in the nonlinear interpolation of a discontinuous function when scaled by a small scale factor. A new nonlinear weight is devised by using an average of the function values and the descaling function, providing the new WCNS schemes (WCNS-Zm/Dm) with many attractive properties. The ENO-property, Si-property and Cp-property of the new WCNS schemes are validated numerically. Results show that the WCNS-Zm/Dm schemes satisfy the ENO-property and Si-property, while only the WCNS-Dm scheme satisfies the Cp-property. In addition, the Gaussian wave problem is solved by using successively refined grids to verify that the optimal order of accuracy of the new schemes can be achieved. Several one-dimensional shock tube problems, and two-dimensional double Mach reflection (DMR) problem and the Riemann IVP problem are simulated to illustrate the ENO-property and Si-property of the scale-invariant WCNS-Zm/Dm schemes.

In this paper, we construct and analyze a Crank-Nicolson fitted finite volume scheme for pricing European options under regime-switching Kou's jump-diffusion model which is governed by a system of partial integro-differential equations (PIDEs). We show that this scheme is consistent, stable and monotone as the mesh sizes in space and time approach zero, hence it ensures the convergence to the solution of continuous problem. Finally, numerical experiments are performed to demonstrate the efficiency, accuracy and robustness of the proposed method.

The study of impacts of down-up hill road segment on the density threshold of traffic shock formation in ring road vehicular flow is helpful to the deep understanding of sags’ bottleneck effect. Sags are freeway segments along which the gradient increases gradually in the traffic direction. The main aim of this paper is to seek the density threshold of shock formation of vehicular flow in ring road with down-up hill segment, because down-up hill roadway segment is a source to cause capacity reduction that is an attractive topic in vehicular traffic science. To seek the density threshold numerically, a viscoelastic continuum model [1] is extended and used. To solve the model equations, a fifth-order weighted essentially non-oscillatory scheme for spatial discretization, and a 3rd order Runge-Kutta scheme for time partial derivative term are used. Validation by existing observation data and the Navier-Stokes like model [2] extended as EZM is done before conducting extensive numerical simulations. For ring road vehicular flow with three separated down-up hill segments, it is found that the density threshold of shock formation decreases monotonically with the relative difference of free flow speed, this variation can be simply fitted by a third order polynomial.

This paper aims to present a new well-balanced, accurate and fast finite volume scheme on unstructured grids to solve hyperbolic conservation laws. It is a scheme that combines both finite volume approach and characteristic method. In this study, we consider a shallow water system with Coriolis effect and bottom friction stresses where this new Finite Volume Characteristics (FVC) scheme has been applied. The physical and mathematical properties of the system, including the C-property, have been well preserved.
First, we developed this approach by preserving the advantages of the finite volume discretization such as conservation property and the method of characteristics, in order to avoid Riemann solvers and to enhance the accuracy without any complexity of the MUSCL reconstruction. Afterward, a discretization was applied to the bottom source term that leads to a well-balanced scheme satisfying the steady-state condition of still water. A semi-implicit treatment will also be presented in this study to avoid stability problems due to source terms. Finally, the proposed finite volume method is verified on several benchmark tests and shows good agreement with analytical solutions and experimental results; moreover, it gives a noteworthy accuracy and rapidity improvement compared to the original approaches.