For many decades meshfree mehtods have been widely studied and developed since they were introduced for the first time in the field of numerical simulation methods. In this talk, we will present a meshfree, particle-based method for Convection-Diffusion problems. It is well known that standard approaches in the framework of finite differences, finite volumes, finite elements, and particle methods work well for elliptical differential equations. However, when applied without the use of a stabilization method to other classes of differential equations, they were found to yield unstable solutions, i.e. solutions that exhibit non-physical spatial oscillations. In our case, the discretization is performed by using piecewise constant kernels and the stabilized scheme we found, is based on a new upwind kernel. We will also show that accurate and stable schemes can be obtained by using special purpose-built kernels, based on a popular stabilization parameter from the finite difference method. In addition, we will establish that under some ideal conditions, the classical optimal finite difference scheme can be derived from the new method. Several numerical tests on four standard problems show the efficiency of the proposed method. Future developments and improvements will also be discussed.
It is known that the semi-discrete ordinary differential equation (ODE) system resulting from spatial discretization of a parabolic partial differential equation, for instance, the heat equation, is highly stiff. Therefore numerical methods with stiff stability such as implicit Runge-Kutta methods and implicit multistep methods are preferred to solve the ODE system. However those methods are usually computationally expensive, especially for nonlinear problems. Rosenbrock method, a special subclass of implicit Runge-Kutta method, is efficient since it is iteration-free for nonlinear problem, but suffers order reduction, when it is applied to non-linear parabolic problems. In this paper we constructed a fourth-order Rosenbrock method to solve the semi-linear parabolic partial differential equation in 1-D with Dirichlet and Neumann boundary conditions. It has been shown that the Rosenbrock method is strongly A-stable hence is suitable for the stiff ODE system obtained from compact finite difference discretization of the reaction-diffusion equation. We have also shown that the new method is free of order reduction when it is applied to nonlinear parabolic problem. Several numerical examples have been solved to demonstrate the efficiency, stability and accuracy of the numerical method.
The Stefan-Maxwell equations form a system of nonlinear partial differential equations (PDE), that describe the diffusion of multiple reacting or non-reacting species in a container. These equations are of particular interest for their applications to biology and chemical engineering. The Stefan-Maxwell equations provide the fluxes of all species in an implicit way as a function of the gradient of partial pressures. The diffusion coefficients depend on the partial fractions of the species,this leads to a system of coupled nonlinear PDEs. In the engineering literature this system is inverted to express each flux as a combination of the gradient of partial fractions before any numerical method is applied. In our talk, we show that the Stefan-Maxwell equations naturally lead to a mixed variational formulation, making the inversion of the system unnecessary before the application of the finite element method. The use of such formalism has not been proposed so far in the literature on Stefan-Maxwell equations. We then show that the nonlinearity can be dealt with through a linearization and conditions for the well posedness of the linearized formulation can be determined. Next, the variational formulation is approximated using mixed finite element methods, in particular with Raviart-Thomas elements. A priori error estimates for the mixed finite elements can be extended to the linearized Stefan-Maxwell equations. Numerical results are presented showing the accuracy of the method with order of convergence matching the error estimates.
We present a semi-implicit scheme for the solution of a PDE that describes the anisotropic propagation of forest fires in heterogeneous landscapes. The goal is to use the scheme to understand the effect of small scales perturbations on the propagation of the front at large scales. One approach is to use the scheme to numerically homogenize the small scale perturbations. Another approach is to generate burn probability maps obtained from large ensembles of realizations for the front arrival time when perturbing the data.