By using the Feynman-Kac formula and combining with Itô-Taylor expansion and finite difference approximation, we first develop an explicit third order one-step method for solving decoupled forward backward stochastic differential equations. Then based on the third order one, an explicit fourth order method is further proposed. Several numerical tests are also presented to illustrate the stability and high order accuracy of the proposed methods.

A second order accurate (in time) numerical scheme is analyzed for the slope-selection (SS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. To make the numerical scheme linear while preserving the nonlinear energy stability, we make use of the scalar auxiliary variable (SAV) approach, in which a modified Crank-Nicolson is applied for the surface diffusion part. The energy stability could be derived a modified form, in comparison with the standard Crank-Nicolson approximation to the surface diffusion term. Such an energy stability leads to an $H^2$ bound for the numerical solution. In addition, this $H^2$ bound is not sufficient for the optimal rate convergence analysis, and we establish a uniform-in-time $H^3$ bound for the numerical solution, based on the higher order Sobolev norm estimate, combined with repeated applications of discrete Hölder inequality and nonlinear embeddings in the Fourier pseudo-spectral space. This discrete $H^3$ bound for the numerical solution enables us to derive the optimal rate error estimate for this alternate SAV method. A few numerical experiments are also presented, which confirm the efficiency and accuracy of the proposed scheme.

In this paper, several new energy identities of metamaterial Maxwell's equations with the perfectly electric conducting (PEC) boundary condition are proposed and proved. These new energy identities are different from the Poynting theorem. By using these new energy identities, it is proved that the Yee scheme on non-uniform rectangular meshes is stable in the discrete $L^2$ and $H^1$ norms when the Courant-Friedrichs-Lewy (CFL) condition is satisfied. Numerical experiments in two-dimension (2D) and 3D are carried out and confirm our analysis, and the superconvergence in the discrete $H^1$ norm is found.

One goal of financial research is to determine fair prices on the financial market. As financial models and the data sets on which they are based are becoming ever larger and thus more complex, financial instruments must be further developed to adapt to the new complexity, with short runtimes and efficient use of memory space. Here we show the effects of combining known strategies and incorporating new ideas to further improve numerical techniques in computational finance.

In this paper we combine an ADI (alternating direction implicit) scheme for the temporal discretization with a sparse grid approach and the combination technique. The later approach considerably reduces the number of "spatial" grid points. The presented standard financial problem for the valuation of American options using the Heston model is chosen to illustrate the advantages of our approach, since it can easily be adapted to other more complex models.

In this paper, a high-accuracy numerical scheme is developed for long-time dynamic simulations of 2D and 3D wave propagation phenomena. In the derivation of the present approach, the second order time derivative of the physical quantity in the wave equation is treated as a substitution variable. Based on the temporal discretization with the Krylov deferred correction (KDC) technique, the original wave problem is then converted into the modified Helmholtz equation. The transformed boundary value problem (BVP) in space is efficiently simulated by using the meshless generalized finite difference method (GFDM) with Taylor series after truncating the second and fourth order approximations. The developed scheme is finally verified by four numerical experiments including cases with complicated domains or the temporally piecewise defined source function. Numerical results match with the analytical solutions and results by the COMSOL software, which demonstrates that the developed KDC-GFDM can allow large time-step sizes for wave propagation problems in long-time intervals.

The study on the fingering phenomenon has been playing an important role in understanding the mechanism of the fluid flow through the porous media. In this paper, a numerical method consisting of the Crank-Nicolson scheme for the temporal discretization and the finite element method for the spatial discretization is proposed for the relaxation non-equilibrium Richards equation in simulating the fingering phenomenon. Towards the efficiency and accuracy of the numerical simulations, a predictor-corrector process is used for resolving the nonlinearity of the equation, and an $h$-adaptive mesh method is introduced for accurately resolving the solution around the wetting front region, in which a heuristic a posteriori error indicator is designed for the purpose. In numerical simulations, a traveling wave solution of the governing equation is derived for checking the numerical convergence of the proposed method. The effectiveness of the $h$-adaptive method is also successfully demonstrated by numerical experiments. Finally the mechanism on generating fingers is discussed by numerically studying several examples.

For flow simulations with complex geometries and structured grids, it is preferred for high-order difference schemes to achieve high accuracy. In order to achieve this goal, the satisfaction of free-stream preservation is necessary to reduce the numerical error in the numerical evaluation of grid metrics. For the linear upwind schemes with flux splitting the free-stream preserving property has been achieved in the early study [Q. Li et al., Commun. Comput. Phys., 22 (2017), pp. 64-94]. In the current paper, new series of nonlinear upwind-biased schemes through WENO interpolation will be proposed. In the new nonlinear schemes, the shock-capturing capability on distorted grids is achieved, which is unavailable for the aforementioned linear upwind schemes. By the inclusion of fluxes on the midpoints, the nonlinearity in the scheme is obtained through WENO interpolations, and the upwind-biased construction is acquired by choosing relevant grid stencils. New third- and fifth-order nonlinear schemes are developed and tested. Discussions are made among proposed schemes, alternative formulations of WENO and hybrid WCNS, in which a general formulation of center scheme with midpoint and nodes employed is obtained as a byproduct. Through the numerical tests, the proposed schemes can achieve the designed orders of accuracy and free-stream preservation property. In 1-D Sod and Shu-Osher problems, the results are consistent with the theoretical predictions. In 2-D cases, the vortex preservation, supersonic inviscid flow around cylinder at $M_\infty=4$, Riemann problem, and shock-vortex interaction problems have been tested. More specifically, two types of grids are employed, i.e., the uniform/smooth grids and the randomized/locally-randomized grids. All schemes can get satisfactory results in uniform/smooth grids. On the randomized grids, most schemes have accomplished computations with reasonable accuracy, except the failure of one third-order scheme in Riemann problem and shock-vortex interaction. Overall, the proposed nonlinear schemes have the capability to solve flow problems on badly deformed grids, and the schemes can be used in the engineering applications.

The study adopts the variational method for analyzing the cantilever tapered beams under a tip load as well as a definite end displacement, and further determining the optimized shapes and materials that can minimize the weights. Two types of beams are taken into account, i.e., the Euler-Bernoulli beam without considering shear deformation and the Timoshenko beam with shear deformation. By using the energy theorem and the reference of isoperimetric problem, the width variation curves and the corresponding minimum masses are derived for both beam types. The optimized curve of beam width for the Euler-Bernoulli beam is found to be a linear function, but nonlinear for the Timoshenko beam. It is also found that the optimized curve in the Timoshenko beam case starts from non-zero at the tip end, but its tendency gradually approaches the one of the Euler-Bernoulli beam. The results indicate that with the increase of the Poisson's ratio, the required minimum mass of the beam will increase no matter how the material changes, suggesting that the optimized mass for the case of Euler-Bernoulli beam is the lower boundary limit which the Timoshenko case cannot go beyond. Furthermore, the ratio $\rho/E$ (density against Elastic Modulus) of the material should be as small as possible, while the ratio $h^2/L^4$ of the beam should be as large as possible in order to minimize the mass for the case of Euler-Bernoulli beam, of which the conclusion is extended to be applicable for the case of Timoshenko beam. In addition, the optimized curves for Euler-Bernoulli beam types are all found to be power functions of length for both tip point load cases and uniform load cases.

Based on fully overlapping domain decomposition, a parallel finite element algorithm for the unsteady Oseen equations is proposed and analyzed. In this algorithm, each processor independently computes a finite element approximate solution in its own subdomain by using a locally refined multiscale mesh at each time step, where conforming finite element pairs are used for the spatial discretizations and backward Euler scheme is used for the temporal discretizations, respectively. Each subproblem is defined in the entire domain with vast majority of the degrees of freedom associated with the particular subdomain that it is responsible for and hence can be solved in parallel with other subproblems using an existing sequential solver without extensive recoding. The algorithm is easy to implement and has low communication cost. Error bounds of the parallel finite element approximate solutions are estimated. Numerical experiments are also given to demonstrate the effectiveness of the algorithm.

A localized version of the method of fundamental solution (LMFS) is devised in this paper for the numerical solutions of three-dimensional (3D) elasticity problems. The present method combines the advantages of high computational efficiency of localized discretization schemes and the pseudo-spectral convergence rate of the classical MFS formulation. Such a combination will be an important improvement to the classical MFS for complicated and large-scale engineering simulations. Numerical examples with up to 100,000 unknowns can be solved without any difficulty on a personal computer using the developed methodologies. The advantages, disadvantages and potential applications of the proposed method, as compared with the classical MFS and boundary element method (BEM), are discussed.

In this paper, a novel symplectic conservative perturbation series expansion method is proposed to investigate the dynamic response of linear Hamiltonian systems accounting for perturbations, which mainly originate from parameters dispersions and measurement errors. Taking the perturbations into account, the perturbed system is regarded as a modification of the nominal system. By combining the perturbation series expansion method with the deterministic linear Hamiltonian system, the solution to the perturbed system is expressed in the form of asymptotic series by introducing a small parameter and a series of Hamiltonian canonical equations to predict the dynamic response are derived. Eventually, the response of the perturbed system can be obtained successfully by solving these Hamiltonian canonical equations using the symplectic difference schemes. The symplectic conservation of the proposed method is demonstrated mathematically indicating that the proposed method can preserve the characteristic property of the system. The performance of the proposed method is evaluated by three examples compared with the Runge-Kutta algorithm. Numerical examples illustrate the superiority of the proposed method in accuracy and stability, especially symplectic conservation for solving linear Hamiltonian systems with perturbations and the applicability in structural dynamic response estimation.

Consider the inverse scattering problem in terms of Helmholtz equation. We study a simply connected domain with oblique derivative boundary condition. In the case of constant $\lambda$, given a finite number of incident wave, the shape of the scatterer is reconstructed from the measured far-field data. We propose a Newton method which is based on the nonlinear boundary integral equation. After computing the Fréchet derivatives with respect to the unknown boundary, the nonlinear equation is transformed to its linear form, then we show the iteration scheme for the inverse problem. We conclude our paper by presenting several numerical examples for shape reconstruction to show the validity of the method we presented.