In this short note we address the issue of numerical resolution and efficiency of high order weig... more In this short note we address the issue of numerical resolution and efficiency of high order weighted essentially nonoscillatory (WENO) schemes for computing solutions containing both discontinuities and complex solution features, through two representative numerical examples: the double Mach reflection problem and the Rayleigh-Taylor instability problem. We conclude that for such solutions with both discontinuities and complex solution features, it is more economical in CPU time to use higher order WENO schemes to obtain comparable numerical resolution.
In this paper, we are concerned with the study of efficient and high order accurate numerical met... more In this paper, we are concerned with the study of efficient and high order accurate numerical methods for solving Hamilton-Jacobi equations with initial conditions defined in the whole domain. One of the commonly used strategy is to solve the problem only in a finite domain, but the determination of boundary conditions at the artificial boundary of the finite computational domain is a problem. If the initial condition decays fast in space, one could use zero boundary condition at the artificial boundary if the domain is large enough, but this may not be very efficient since the computational domain may need to be very large to justify this choice. In this paper we use the high order moving mesh arbitrary Lagrangian Eulerian (ALE) weighted essentially non-oscillatory (WENO) finite difference scheme, recently developed in , in a finite and moving computational domain, with numerical boundary conditions obtained by solving the characteristic ordinary differential equations (ODEs) along the artificial boundary of the moving computational domain. This method works well when singularities do not appear at the artificial boundary. Extensive numerical tests in one and two dimensions are given to demonstrate the flexibility and efficiency of our method in solving both smooth problems and problems with corner singularities.
ESAIM: Mathematical Modelling and Numerical Analysis, 2019
In this paper, we discuss the stability and error estimates of the fully discrete schemes for lin... more In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed f...
Computer Methods in Applied Mechanics and Engineering, 2017
The purpose of this work is to incorporate numerically, in a discontinuous Galerkin (DG) solver o... more The purpose of this work is to incorporate numerically, in a discontinuous Galerkin (DG) solver of a Boltzmann-Poisson model for hot electron transport, an electronic conduction band whose values are obtained by the spherical averaging of the full band structure given by a local empirical pseudopotential method (EPM) around a local minimum of the conduction band for silicon, as a midpoint between a radial band model and an anisotropic full band, in order to provide a more accurate physical description of the electron group velocity and conduction energy band structure in a semiconductor. This gives a better quantitative description of the transport and collision phenomena that fundamentally define the behaviour of the Boltzmann -Poisson model for electron transport used in this work. The numerical values of the derivatives of this conduction energy band, needed for the description of the electron group velocity, are obtained by means of a cubic spline interpolation. The EPM-Boltzmann-Poisson transport with this spherically averaged EPM calculated energy surface is numerically simulated and compared to the output of traditional analytic band models such as the parabolic and Kane bands, numerically implemented too, for the case of 1D n + -n -n + silicon diodes with 400nm and 50nm channels. Quantitative differences are observed in the kinetic moments related to the conduction energy band used, such as mean velocity, average energy, and electric current (momentum).
Fixed-point iterative sweeping methods were developed in the literature to efficiently solve stat... more Fixed-point iterative sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilizes the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. They take advantage of the properties of hyperbolic partial differential equations (PDEs) and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping order. Different from other fast sweeping methods, fixed-point iterative sweeping methods have the advantages such as that they have explicit forms and do not involve inverse operation of nonlinear local systems. In principle, it can be applied in solving very general equations using any monotone numerical fluxes and high order approximations easily. In this paper, based on the recently developed fifth order WENO schemes which improve the convergence of the classical WENO schemes by removing slight pos...
Wave propagation problems arise in a wide range of applications. The energy conserving property i... more Wave propagation problems arise in a wide range of applications. The energy conserving property is one of the guiding principles for numerical algorithms, in order to minimize the phase or shape errors after long time integration. In this paper, we develop and analyze a local discontinuous Galerkin (LDG) method for solving the wave equation. We prove optimal error estimates, superconvergence toward a particular projection of the exact solution, and the energy conserving property for the semi-discrete formulation. The analysis is extended to the fully discrete LDG scheme, with the centered second-order time discretization (the leap-frog scheme). Our numerical experiments demonstrate optimal rates of convergence and superconvergence. We also show that the shape of the solution, after long time integration, is well preserved due to the energy conserving property.
Simulation of Semiconductor Processes and Devices 2007
We present preliminary results of a discontinuous Galerkin scheme applied to deterministic comput... more We present preliminary results of a discontinuous Galerkin scheme applied to deterministic computations of the transients for the Boltzmann-Poisson system describing electron transport in semiconductor devices. The collisional term models optical-phonon interactions which become dominant under strong energetic conditions corresponding to nanoscale active regions under applied bias. The proposed numerical technique, that is a finite element method which uses discontinuous piecewise polynomials as basis functions, is applied for investigating the carrier transport in bulk silicon and in a silicon n + -nn + diode. Additionally, the obtained results are compared to those of a high order WENO scheme solver.
This note aims at demonstrating the advantage of moving-water well-balanced schemes over still-wa... more This note aims at demonstrating the advantage of moving-water well-balanced schemes over still-water well-balanced schemes for the shallow water equations. We concentrate on numerical examples with solutions near a moving-water equilibrium. For such examples, still-water well-balanced methods are not capable of capturing the small perturbations of the moving-water equilibrium and may generate significant spurious oscillations, unless an extremely refined mesh is used. On the other hand, movingwater well-balanced methods perform well in these tests. The numerical examples in this note clearly demonstrate the importance of utilizing moving-water well-balanced methods for solutions near a moving-water equilibrium.
In this paper, we develop a local discontinuous Galerkin (LDG) finite element method for surface ... more In this paper, we develop a local discontinuous Galerkin (LDG) finite element method for surface diffusion and Willmore flow of graphs. We prove L 2 stability for the equation of surface diffusion of graphs and energy stability for the equation of Willmore flow of graphs. We provide numerical simulation results for different types of solutions of these two types of the equations to illustrate the accuracy and capability of the LDG method.
In this short note we address the issue of numerical resolution and efficiency of high order weig... more In this short note we address the issue of numerical resolution and efficiency of high order weighted essentially nonoscillatory (WENO) schemes for computing solutions containing both discontinuities and complex solution features, through two representative numerical examples: the double Mach reflection problem and the Rayleigh-Taylor instability problem. We conclude that for such solutions with both discontinuities and complex solution features, it is more economical in CPU time to use higher order WENO schemes to obtain comparable numerical resolution.
In this paper, we are concerned with the study of efficient and high order accurate numerical met... more In this paper, we are concerned with the study of efficient and high order accurate numerical methods for solving Hamilton-Jacobi equations with initial conditions defined in the whole domain. One of the commonly used strategy is to solve the problem only in a finite domain, but the determination of boundary conditions at the artificial boundary of the finite computational domain is a problem. If the initial condition decays fast in space, one could use zero boundary condition at the artificial boundary if the domain is large enough, but this may not be very efficient since the computational domain may need to be very large to justify this choice. In this paper we use the high order moving mesh arbitrary Lagrangian Eulerian (ALE) weighted essentially non-oscillatory (WENO) finite difference scheme, recently developed in , in a finite and moving computational domain, with numerical boundary conditions obtained by solving the characteristic ordinary differential equations (ODEs) along the artificial boundary of the moving computational domain. This method works well when singularities do not appear at the artificial boundary. Extensive numerical tests in one and two dimensions are given to demonstrate the flexibility and efficiency of our method in solving both smooth problems and problems with corner singularities.
ESAIM: Mathematical Modelling and Numerical Analysis, 2019
In this paper, we discuss the stability and error estimates of the fully discrete schemes for lin... more In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed f...
Computer Methods in Applied Mechanics and Engineering, 2017
The purpose of this work is to incorporate numerically, in a discontinuous Galerkin (DG) solver o... more The purpose of this work is to incorporate numerically, in a discontinuous Galerkin (DG) solver of a Boltzmann-Poisson model for hot electron transport, an electronic conduction band whose values are obtained by the spherical averaging of the full band structure given by a local empirical pseudopotential method (EPM) around a local minimum of the conduction band for silicon, as a midpoint between a radial band model and an anisotropic full band, in order to provide a more accurate physical description of the electron group velocity and conduction energy band structure in a semiconductor. This gives a better quantitative description of the transport and collision phenomena that fundamentally define the behaviour of the Boltzmann -Poisson model for electron transport used in this work. The numerical values of the derivatives of this conduction energy band, needed for the description of the electron group velocity, are obtained by means of a cubic spline interpolation. The EPM-Boltzmann-Poisson transport with this spherically averaged EPM calculated energy surface is numerically simulated and compared to the output of traditional analytic band models such as the parabolic and Kane bands, numerically implemented too, for the case of 1D n + -n -n + silicon diodes with 400nm and 50nm channels. Quantitative differences are observed in the kinetic moments related to the conduction energy band used, such as mean velocity, average energy, and electric current (momentum).
Fixed-point iterative sweeping methods were developed in the literature to efficiently solve stat... more Fixed-point iterative sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilizes the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. They take advantage of the properties of hyperbolic partial differential equations (PDEs) and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping order. Different from other fast sweeping methods, fixed-point iterative sweeping methods have the advantages such as that they have explicit forms and do not involve inverse operation of nonlinear local systems. In principle, it can be applied in solving very general equations using any monotone numerical fluxes and high order approximations easily. In this paper, based on the recently developed fifth order WENO schemes which improve the convergence of the classical WENO schemes by removing slight pos...
Wave propagation problems arise in a wide range of applications. The energy conserving property i... more Wave propagation problems arise in a wide range of applications. The energy conserving property is one of the guiding principles for numerical algorithms, in order to minimize the phase or shape errors after long time integration. In this paper, we develop and analyze a local discontinuous Galerkin (LDG) method for solving the wave equation. We prove optimal error estimates, superconvergence toward a particular projection of the exact solution, and the energy conserving property for the semi-discrete formulation. The analysis is extended to the fully discrete LDG scheme, with the centered second-order time discretization (the leap-frog scheme). Our numerical experiments demonstrate optimal rates of convergence and superconvergence. We also show that the shape of the solution, after long time integration, is well preserved due to the energy conserving property.
Simulation of Semiconductor Processes and Devices 2007
We present preliminary results of a discontinuous Galerkin scheme applied to deterministic comput... more We present preliminary results of a discontinuous Galerkin scheme applied to deterministic computations of the transients for the Boltzmann-Poisson system describing electron transport in semiconductor devices. The collisional term models optical-phonon interactions which become dominant under strong energetic conditions corresponding to nanoscale active regions under applied bias. The proposed numerical technique, that is a finite element method which uses discontinuous piecewise polynomials as basis functions, is applied for investigating the carrier transport in bulk silicon and in a silicon n + -nn + diode. Additionally, the obtained results are compared to those of a high order WENO scheme solver.
This note aims at demonstrating the advantage of moving-water well-balanced schemes over still-wa... more This note aims at demonstrating the advantage of moving-water well-balanced schemes over still-water well-balanced schemes for the shallow water equations. We concentrate on numerical examples with solutions near a moving-water equilibrium. For such examples, still-water well-balanced methods are not capable of capturing the small perturbations of the moving-water equilibrium and may generate significant spurious oscillations, unless an extremely refined mesh is used. On the other hand, movingwater well-balanced methods perform well in these tests. The numerical examples in this note clearly demonstrate the importance of utilizing moving-water well-balanced methods for solutions near a moving-water equilibrium.
In this paper, we develop a local discontinuous Galerkin (LDG) finite element method for surface ... more In this paper, we develop a local discontinuous Galerkin (LDG) finite element method for surface diffusion and Willmore flow of graphs. We prove L 2 stability for the equation of surface diffusion of graphs and energy stability for the equation of Willmore flow of graphs. We provide numerical simulation results for different types of solutions of these two types of the equations to illustrate the accuracy and capability of the LDG method.
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