A Second Order Accurate in Time, Energy Stable Finite Element Scheme for the Flory-Huggins-Cahn-Hilliard Equation

In this paper, we propose and analyze a second order accurate in time, masslumped mixed finite element scheme for the Cahn-Hilliard equation with a logarithmic Flory-Huggins energy potential. The standard backward differentiation formula (BDF)stencil is applied in the temporal discretization. In the chemical potential approximation,both the logarithmic singular terms and the surface diffusion term are treatedimplicitly, while the expansive term is explicitly updated via a second-order Adams-Bashforth extrapolation formula, following the idea of the convex-concave decompositionof the energy functional. In addition, an artificial Douglas-Dupont regularizationterm is added to ensure the energy dissipativity. In the spatial discretization, the masslumped finite element method is adopted. We provide a theoretical justification of theunique solvability of the mass lumped finite element scheme, using a piecewise linearelement. In particular, the positivity is always preserved for the logarithmic argumentsin the sense that the phase variable is always located between -1 and 1. In fact, thesingular nature of the implicit terms and the mass lumped approach play an essentialrole in the positivity preservation in the discrete setting. Subsequently, an unconditionalenergy stability is proven for the proposed numerical scheme. In addition, theconvergence analysis and error estimate of the numerical scheme are also presented.Two numerical experiments are carried out to verify the theoretical properties.

Positive and Conservative Characteristic Block-Centered Finite Difference Methods for Convection Dominated Diffusion Equations

In this work,spatial second order positivity preserving characteristic blockcentered finite difference methods are proposed for solving convection dominated diffusion problems.By using a conservative piecewise parabolic interpolation with positive constraint,the temporal first order scheme is shown to conserve mass exactly and preserve the positivity property of solution.Taking advantage of characteristics,there is no strict restriction on time steps.The scheme is extended to temporal second order by using a particular extrapolation along the characteristics.To restore solution positivity,a mass conservative local limiter is introduced and verified to keep second order accuracy.Numerical examples are carried out to demonstrate the performance of proposed methods.

GPU Accelerated Discontinuous Galerkin Methods for Shallow Water Equations

We discuss the development,verification,and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations.The shallow water equations are hyperbolic partial differential equations and are widely used in the simulation of tsunami wave propagations.Our algorithms are tailored to take advantage of the single instruction multiple data(SIMD)architecture of graphic processing units.The time integration is accelerated by local time stepping based on a multi-rate Adams-Bashforth scheme.A total variational bounded limiter is adopted for nonlinear stability of the numerical scheme.This limiter is coupled with a mass and momentum conserving positivity preserving limiter for the special treatment of a dry or partially wet element in the triangulation.Accuracy,robustness and performance are demonstrated with the aid of test cases.Furthermore,we developed a unified multi-threading model OCCA.The kernels expressed in OCCA model can be cross-compiled with multi-threading models OpenCL,CUDA,and OpenMP.We compare the performance of the OCCA kernels when cross-compiled with these models.

A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model

In this paper we propose and analyze a second order accurate numericalscheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order AdamsBashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme,which ensures the positivity-preserving property, i.e., the numerical value of the phasevariable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special formof the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearizedstability analysis. A few numerical results, including both the constant-mobility andsolution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.

Exponential Runge-Kutta Methods for the Multispecies Boltzmann Equation

This paper generalizes the exponential Runge-Kutta asymptotic preserving(AP)method developed in[G.Dimarco and L.Pareschi,SIAM Numer.Anal.,49(2011),pp.2057–2077]to compute the multi-species Boltzmann equation.Compared to the single species Boltzmann equation that the method was originally applied on,this set of equation presents a new difficulty that comes from the lack of local conservation laws due to the interaction between different species.Hence extra stiff nonlinear source terms need to be treated properly to maintain the accuracy and the AP property.The method we propose does not contain any nonlinear nonlocal implicit solver,and can capture the hydrodynamic limit with time step and mesh size independent of the Knudsen number.We prove the positivity and strong AP properties of the scheme,which are verified by two numerical examples.