Numerical methods for second-order stochastic differential equations

We seek numerical methods for second‐order stochastic differential equations that reproduce the stationary density accurately for all values of damping. A complete analysis is possible for scalar linear second‐order equations (damped harmonic oscillators with additive noise), where the statistics are Gaussian and can be calculated exactly in the continuous‐time and discrete‐time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge–Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case.

Numerical exploration of plastic deformation mechanisms of copper nanowires with surface defects

Based on the molecular dynamics simulation, plastic deformation mechanisms associated with the zigzag stress curves in perfect and surface defected copper nanowires under uniaxial tension are studied. In our previous study, it has found that the surface defect exerts larger influence than the centro-plane defect, and the 45o surface defect appears as the most influential surface defect. Hence, in this paper, the nanowire with a 45o surface defect is chosen to investigate the defect’s effect to the plastic deformation mechanism of nanowires. We find that during the plastic deformation of both perfect and defected nanowires, decrease regions of the stress curve are accompanied with stacking faults generation and migration activities, but during stress increase, the structure of the nanowire appears almost unchanged. We also observe that surface defects have obvious influence on the nanowire’s plastic deformation mechanisms. In particular, only two sets of slip planes are found to be active and twins are also observed in the defected nanowire.

Time-dependent fractional advection–diffusion equations by an implicit MLS meshless method

Recently, many new applications in engineering and science are governed by a series of fractional partial differential equations (FPDEs). Unlike the normal partial differential equations (PDEs), the differential order in a FPDE is with a fractional order, which will lead to new challenges for numerical simulation, because most existing numerical simulation techniques are developed for the PDE with an integer differential order. The current dominant numerical method for FPDEs is Finite Difference Method (FDM), which is usually difficult to handle a complex problem domain, and also hard to use irregular nodal distribution. This paper aims to develop an implicit meshless approach based on the moving least squares (MLS) approximation for numerical simulation of fractional advection-diffusion equations (FADE), which is a typical FPDE. The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless strong-forms. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. Several numerical examples with different problem domains and different nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the FADE.