Stability analysis of a second-order difference scheme for the time-fractional mixed sub-diffusion and diffusion-wave equation

This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0,1). By providing an a priori estimate of the solution, we have established the existence and uniqueness of a numerical solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2 type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings

A generalised finite difference scheme based on compact integrated radial basis function for flow in heterogeneous soils

In the present paper, we develop a generalised finite difference approach based on compact integrated radial basis function (CIRBF) stencils for solving highly nonlinear Richards equation governing fluid movement in heterogeneous soils. The proposed CIRBF scheme enjoys a high level of accuracy and a fast convergence rate with grid refinement owing to the combination of the integrated RBF approximation and compact approximation where the spatial derivatives are discretised in terms of the information of neighbouring nodes in a stencil. The CIRBF method is first verified through the solution of ordinary differential equations, 2-D Poisson equations and a Taylor-Green vortex. Numerical comparisons show that the CIRBF method outperforms some other methods in the literature. The CIRBF method in conjunction with a rational function transformation method and an adaptive time-stepping scheme is then applied to simulate 1-D and 2-D soil infiltrations effectively. The proposed solutions are more accurate and converge faster than those of the finite different method employed with a second-order central difference scheme. Additionally, the present scheme also takes less time to achieve target accuracy in comparison with the 1D-IRBF and HOC schemes

Two-dimensional finite-difference model for moving boundary hydrodynamic problems

To predict the hydrodynamics of lakes, estuaries and shallow seas, a two 'dimensional numerical model is developed using the method of fractional steps. The governing equations, i.e., the vertically integrated Navier-Stokes equations of fluid motion, are solved through three steps: advection, diffusion and propagation. The characteristics method is used to solve the advection, the alternating direction implicit method is applied to compute the diffusion, and the conjugate gradient iterative method is employed to calculate the propagation. Two ways to simulate the moving boundary problem are studied. The first method is based on the weir formulation. The second method is based on the assumption that a thin water layer exists over the entire dry region at all times. A number of analytical solutions are used to validate the model. The model is also applied to simulate the wind driven circulation in Lake Okeechobee, Florida. (135 page document

Development of a fractional-step method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems

A fractional step method is developed for solving the time-dependent three-dimensional incompressible Navier-Stokes equations in generalized coordinate systems. The primitive variable formulation uses the pressure, defined at the center of the computational cell, and the volume fluxes across the faces of the cells as the dependent variables, instead of the Cartesian components of the velocity. This choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell. The governing equations are discretized by finite volumes using a staggered mesh system. The solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a Poisson equation. This procedure, combined with the consistent approximations of the geometric quantities, is done to satisfy the discretized mass conservation equation to machine accuracy, as well as to gain the favorable convergence properties of the Poisson solver. The momentum equations are solved by an approximate factorization method, and a novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two- and three-dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method. Good agreement is obtained in all cases

Process Chain for Numerical Simulation of IMLS

Additive layer manufacturing methods imply, among other advantages, extensive flexibility concerning their ability to realize mass customization. Despite various efforts towards process enhancement, numerous deficiencies concerning part distortion or residual stresses are still observable. The present work deals with the definition of an efficient process chain for numerical simulation of indirect metal laser sintering (IMLS), in order to improve dimensional accuracy. The underlying method is based on investigations of dilatometric behavior of iron based powder, which is integrated into reaction kinetic models and coupled with a finite element analysis (FEA). Thus, singular process steps, e. g. solid phase sintering, phase transformations or infiltration, are numerically modelled with adequate accuracy. Referring to thermomechanical simulation, possibilities for pre-scaling of part geometries are presented.Mechanical Engineerin

Adaptive finite element method assisted by stochastic simulation of chemical systems

Stochastic models of chemical systems are often analysed by solving the corresponding\ud Fokker-Planck equation which is a drift-diffusion partial differential equation for the probability\ud distribution function. Efficient numerical solution of the Fokker-Planck equation requires adaptive mesh refinements. In this paper, we present a mesh refinement approach which makes use of a stochastic simulation of the underlying chemical system. By observing the stochastic trajectory for a relatively short amount of time, the areas of the state space with non-negligible probability density are identified. By refining the finite element mesh in these areas, and coarsening elsewhere, a suitable mesh is constructed and used for the computation of the probability density