Search

In the continuous application and development of network technology, the reform in Internet information will inevitably bring about major changes in college education and teaching. Based on this background, the paper puts forward the influence of ‘Internet+’ on art design education. At the same time, the article explores the influence factors of the Internet on art design teaching based on the non-linear equation model. We use the parameterised cubic clipping algorithm to find the roots of non-polynomial equations. Through performance analysis and comparison, it is found that the root-finding algorithm proposed by us has better parallelism and lower-storage requirements. This algorithm has certain advantages in solving the weight of the factors that influence the Internet to art design teaching.

We construct, analyze, and numerically validate a class of conservative discontinuous Galerkin (DG) schemes for the Schrodinger-Poisson equation. The proposed schemes all shown to conserve both mass and energy. For the semi-discrete DG scheme the optimal L2 error estimates are provided. Efficient iterative algorithms are also constructed to solve the second-order implicit time discretization. The presented numerical tests demonstrate the method’s accuracy and robustness, confirming that the conservation properties help to reproduce faithful solutions over long time simulation.

We provide a framework for the analysis of the direct discontinuous Galerkin (DDG) methods for multi-dimensional convection–diffusion equations subject to various boundary conditions. A key tool is the global projection constructed by the DDG scheme applied to an associated elliptic problem. Such projection is well-defined for a class of diffusive flux parameters, and the optimal projection error in $$L^2$$ is obtained with an arbitrary locally regular partition of the domain and for an arbitrary degree of polynomials. This results in the optimal $$L^2$$ error for the DDG method to the elliptic problem, and further leading to the optimal $$L^2$$ error for the DDG method to the convection–diffusion problem.

A dynamic mass-transport method is proposed for approximately solving the Poisson-Nernst-Planck(PNP) equations. The semi-discrete scheme based on the JKO type variational formulation naturally enforces solution positivity and the energy law as for the continuous PNP system. The fully discrete scheme is further formulated as a constrained minimization problem, shown to be solvable, and satisfy all three solution properties (mass conservation, positivity and energy dissipation) independent of time step size or the spatial mesh size. Numerical experiments are conducted to validate convergence of the computed solutions and verify the structure preserving property of the proposed scheme.

We develop a new finite difference scheme for the Maxwell-Stefan diffusion system. The scheme is conservative, energy stable and positivity-preserving. These nice properties stem from a variational structure and are proved by reformulating the finite difference scheme into an equivalent optimization problem. The solution to the scheme emerges as the minimizer of the optimization problem, and as a consequence energy stability and positivity-preserving properties are obtained.

This paper presents a partial differential equation framework for deep residual neural networks and for the associated learning problem. This is done by carrying out the continuum limits of neural networks with respect to width and depth. We study the wellposedness, the large time solution behavior, and the characterization of the steady states of the forward problem. Several useful time-uniform estimates and stability/instability conditions are presented. We state and prove optimality conditions for the inverse deep learning problem, using standard variational calculus, the Hamilton-Jacobi-Bellmann equation and the Pontryagin maximum principle. This serves to establish a mathematical foundation for investigating the algorithmic and theoretical connections between neural networks, PDE theory, variational analysis, optimal control, and deep learning.

This paper is concerned with the critical threshold phenomenon for one-dimensional damped, pressureless Euler–Poisson equations with electric force induced by a constant background, originally studied in [S. Engelberg and H. Liu and E. Tadmor, Indiana Univ. Math. J. 50 (2001) 109–157]. A simple transformation is used to linearize the characteristic system of equations, which allows us to study the geometrical structure of critical threshold curves for three damping cases: overdamped, underdamped and borderline damped through phase plane analysis. We also derive the explicit form of these critical curves. These sharp results state that if the initial data is within the threshold region, the solution will remain smooth for all time, otherwise it will have a finite time breakdown. Finally, we apply these general results to identify critical thresholds for a non-local system subjected to initial data on the whole line.

This paper is concerned with large time behavior of solutions to a semi-discrete model involving nonlinear competition that describes the evolution of a trait-structured population. Under some threshold assumptions, the steady solution is shown unique and strictly positive, and also globally stable. The exponential convergence rate to the steady state is also established. These results are consistent with the results in [P.-E. Jabin, H. L. Liu. Nonlinearity 30 (2017) 4220–4238] for the continuous model.

For a class of convection-diffusion equations with variable diffusivity, we construct third order accurate discontinuous Galerkin (DG) schemes on both one and two dimensional rectangular meshes. The DG method with an explicit time stepping can well be applied to nonlinear convection–diffusion equations. It is shown that under suitable time step restrictions, the scaling limiter proposed in Liu and Yu (2014) [23] when coupled with the present DG schemes preserves the solution bounds indicated by the initial data, i.e., the maximum principle, while maintaining uniform third order accuracy. These schemes can be extended to rectangular meshes in three dimension. The crucial for all model scenarios is that an effective test set can be identified to verify the desired bounds of numerical solutions. This is achieved mainly by taking advantage of the flexible form of the diffusive flux and the adaptable decomposition of weighted cell averages. Numerical results are presented to validate the numerical methods and demonstrate their effectiveness.

We design, analyze and numerically validate alternating evolution schemes for solving static Hamilton–Jacobi equations. This class of high resolution numerical schemes is based on the alternating evolution reformulation of the original Hamilton–Jacobi equation. Stability and convergence properties are proven for first order schemes. Numerical implementation of up to third order schemes is carried out for a set of widely used examples. Further application of the proposed method to the kinetic Hamilton–Jacobi equation is given, while the Hamiltonian is determined by an integral in phase space.

In this paper, we design, analyze, and numerically validate positive and energy-dissipating schemes for solving the time-dependent multi-dimensional system of Poisson–Nernst–Planck equations, which has found much use in the modeling of biological membrane channels and semiconductor devices. The semi-implicit time discretization based on a reformulation of the system gives a well-posed elliptic system, which is shown to preserve solution positivity for arbitrary time steps. The first order (in time) fully-discrete scheme is shown to preserve solution positivity and mass conservation unconditionally, and energy dissipation with only a mild O(1) time step restriction. The scheme is also shown to preserve the steady-states. For the fully second order (in both time and space) scheme with large time steps, solution positivity is restored by a local scaling limiter, which is shown to maintain the spatial accuracy. These schemes are easy to implement. Several three-dimensional numerical examples verify our theoretical findings and demonstrate the accuracy, efficiency, and robustness of the proposed schemes, as well as the fast approach to steady-states.

In this paper, we design and analyze third order positivity-preserving discontinuous Galerkin (DG) schemes for solving the time-dependent system of Poisson--Nernst--Planck (PNP) equations, which has found much use in diverse applications. Our DG method with Euler forward time discretization is shown to preserve the positivity of cell averages at all time steps. The positivity of numerical solutions is then restored by a scaling limiter in reference to positive weighted cell averages. The method is also shown to preserve steady states. Numerical examples are presented to demonstrate the third order accuracy and illustrate the positivity-preserving property in both one and two dimensions., 7 figures, 16 tables

We propose and study a one-dimensional $2\times 2$ hyperbolic Eulerian system with local relaxation from critical threshold phenomena perspective. The system features dynamic transition between strictly and weakly hyperbolic. For different classes of relaxation we identify intrinsic critical thresholds for initial data that distinguish global regularity and finite time blowup. For relaxation independent of density, we estimate bounds on density in terms of velocity where the system is strictly hyperbolic., 19 pages, 2 figures

We present a data-driven optimal control approach which integrates the reported partial data with the epidemic dynamics for COVID-19. We use a basic Susceptible-Exposed-Infectious-Recovered (SEIR) model, the model parameters are time-varying and learned from the data. This approach serves to forecast the evolution of the outbreak over a relatively short time period and provide scheduled controls of the epidemic. We provide efficient numerical algorithms based on a generalized Pontryagin Maximum Principle associated with the optimal control theory. Numerical experiments demonstrate the effective performance of the proposed model and its numerical approximations., 20 pages, 5 figures

This paper is concerned with a mathematical model of competition for resource where species consume noninteracting resources. This system of differential equations is formally obtained by renormalizing the MacArthur's competition model at equilibrium, and agrees with the trait-continuous model studied by Mirrahimi S, Perthame B, Wakano JY [J. Math. Biol. 64(7): 1189-1223, 2012]. As a dynamical system, self-organized generation of distinct species occurs. The necessary conditions for survival are given. We prove the existence of the evolutionary stable distribution (ESD) through an optimization problem and present an independent algorithm to compute the ESD directly. Under certain structural conditions, solutions of the system are shown to approach the discrete ESD as time evolves. The time discretization of the system is proven to satisfy two desired properties: positivity and energy dissipation. Numerical examples are given to illustrate certain interesting biological phenomena.

Many problems in solid state physics and quantum chemistry require the solution of the Schrodinger equation in the presence of an electromagnetic field. In this paper, we construct, analyze, and nu...

In this paper, we present the stability analysis and error estimates for the alternating evolution discontinuous Galerkin (AEDG) method with third order explicit Runge-Kutta temporal discretization for linear convection-diffusion equations. The scheme is shown stable under a CFL-like stability condition c0τ ≤ ε ≤ c1h2. Here ε is the method parameter, and h is the maximum spatial grid size. We further obtain the optimal L2 error of order O(τ3 + hk+1). Key tools include two approximation finite element spaces to distinguish overlapping polynomials, coupled global projections, and energy estimates of errors. For completeness, the stability analysis and error estimates for second order explicit Runge-Kutta temporal discretization is included in the appendix.

A novel discontinuous Galerkin (DG) method is developed to solve time-dependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed formulation and central interface numerical fluxes so that the resulting semi-discrete schemes are $L^2$ stable even without interior penalty. For time discretization, we use Crank-Nicolson so that the resulting scheme is unconditionally stable and second order in time. We present the optimal $L^2$ error estimate of $O(h^{k+1})$ for polynomials of degree $k$ for semi-discrete DG schemes, and the $L^2$ error of $O(h^{k+1} +(\Delta t)^2)$ for fully discrete DG schemes. Extensions to more general fourth order partial differential equations and cases with non-homogeneous boundary conditions are provided. Numerical results are presented to verify the stability and accuracy of the schemes. Finally, an application to the one-dimensional Swift-Hohenberg equation endowed with a decay free energy is presented., Comment: 30 pages, 9 figures

In this paper, we introduce an invariant-region-preserving (IRP) limiter for the p-system and the corresponding viscous p-system, both of which share the same invariant region. Rigorous analysis is presented to show that for smooth solutions the order of approximation accuracy is not destroyed by the IRP limiter, provided the cell average stays away from the boundary of the invariant region. Moreover, this limiter is explicit, and easy for computer implementation. A generic algorithm incorporating the IRP limiter is presented for high order finite volume type schemes as long as the evolved cell average of the underlying scheme stays strictly within the invariant region. For any high order discontinuous Galerkin (DG) scheme to the p-system, sufficient conditions are obtained for cell averages to stay in the invariant region. For the viscous p-system, we design both second and third order IRP DG schemes. Numerical experiments are provided to test the proven properties of the IRP limiter and the performance of IRP DG schemes., 28 pages, 3 figures, 9 tables, accepted for publication in Numerical Methods for Partial Differential Equations, 2018

In this paper, we design, analyze and numerically validate energy dissipating finite volume schemes for a competition-mutation equation with a gradient flow structure. The model describes the evolution of a population structured with respect to a continuous trait. Both semi-discrete and fully discrete schemes are demonstrated to satisfy the two desired properties: positivity of numerical solutions and energy dissipation. These ensure that the positive steady state is asymptotically stable. Moreover, the discrete steady state is proven to be the same as the minimizer of a discrete energy function. As a comparison, the positive steady state can also be produced by a nonlinear programming solver. Finally, a series of numerical tests is provided to demonstrate both accuracy and the energy dissipation property of the numerical schemes. The numerical solutions of the model with small mutation are shown to be close to those of the corresponding model with linear competition.

In this paper, we numerically study a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, where the initial data have an algebraic singularity in vorticity at the origin. These are different from the multi-dimensional Riemann problems widely studied in the literature. Our computations provide numerical evidence of the existence of initial value problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. The compressible Euler equations are solved using the positivity-preserving discontinuous Galerkin method., Comment: 16 pages, 26 figures; accepted manuscript

In this paper, we introduce novel discontinuous Galerkin (DG) schemes for the Cahn-Hilliard equation, which arises in many applications. The method is designed by integrating the mixed DG method for the spatial discretization with the \emph{Invariant Energy Quadratization} (IEQ) approach for the time discretization. Coupled with a spatial projection, the resulting IEQ-DG schemes are shown to be unconditionally energy dissipative, and can be efficiently solved without resorting to any iteration method. Both one and two dimensional numerical examples are provided to verify the theoretical results, and demonstrate the good performance of IEQ-DG in terms of efficiency, accuracy, and preservation of the desired solution properties., 25 pages, 23 figures

The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here we introduce fully discrete discontinuous Galerkin (DG) schemes for a class of fourth order gradient flow problems, including the nonlinear Swift-Hohenberg equation, to produce free-energy-decaying discrete solutions, irrespective of the time step and the mesh size. We exploit and extend the mixed DG method introduced in [H. Liu and P. Yin, J. Sci. Comput., 77: 467--501, 2018] for the spatial discretization, and the "Invariant Energy Quadratization" method for the time discretization. The resulting IEQ-DG algorithms are linear, thus they can be efficiently solved without resorting to any iteration method. We actually prove that these schemes are unconditionally energy stable. We present several numerical examples that support our theoretical results and illustrate the efficiency, accuracy and energy stability of our new algorithm. The numerical results on two dimensional pattern formation problems indicate that the method is able to deliver comparable patterns of high accuracy., 24 pages, 24 figures

Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool to generate more general high frequency solutions to PDEs. We present a superposition of Gaussian beams over an arbitrary bounded set of dimension $m$ in phase space, and show that the tools recently developed in [ H. Liu, O. Runborg, and N. M. Tanushev, Math. Comp., 82: 919--952, 2013] can be applied to obtain the propagation error of order $k^{1- \frac{N}{2}- \frac{d-m}{4}}$, where $N$ is the order of beams and $d$ is the spatial dimension. Moreover, we study the sharpness of this estimate in examples., 24, 1 figure

This paper investigates oscillatory traveling wave solutions to an attractive chemotaxis system. The convective part of this system changes its type when crossing a parabola in the phase space. The oscillatory nature of the traveling wave comes from the fact that one far-field state is in the elliptic region and another in the hyperbolic region. Such traveling wave solutions are shown to be linearly unstable. Detailed construction of some traveling wave solutions is presented.

The invariant preserving property is one of the guiding principles for numerical algorithms in solving wave equations, in order to minimize phase and amplitude errors after long time simulation. In this paper, we design, analyze and numerically validate a Hamiltonian preserving discontinuous Galerkin method for solving the Korteweg-de Vries (KdV) equation. For the generalized KdV equation, the semi-discrete formulation is shown to preserve both the first and the third conserved integrals, and approximately preserve the second conserved integral; for the linearized KdV equation, all the first three conserved integrals are preserved, and optimal error estimates are obtained for polynomials of even degree. The preservation properties are also maintained by the fully discrete DG scheme. Our numerical experiments demonstrate both high accuracy of convergence and preservation of all three conserved integrals for the generalized KdV equation. We also show that the shape of the solution, after long time simulation, is well preserved due to the Hamiltonian preserving property.

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for one-dimensional linear convection-diffusion equations. We prove, under some suitable choice of numerical fluxes and initial discretization, a 2k-th and ( k + 2 ) -th order superconvergence rate of the DDG approximation at nodes and Lobatto points, respectively, and a ( k + 1 ) -th order of the derivative approximation at Gauss points, where k is the polynomial degree. Moreover, we also prove that the DDG solution is superconvergent with an order k + 2 to a particular projection of the exact solution. Numerical experiments are presented to validate the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 290–317, 2017

The purpose of this paper is to develop and test novel invariant-preserving finite difference schemes for both the Camassa-Holm (CH) equation and one of its 2-component generalizations (2CH). The considered PDEs are strongly nonlinear, admitting soliton-like peakon solutions which are characterized by a slope discontinuity at the peak in the wave shape, and therefore suitable for modeling both short wave breaking and long wave propagation phenomena. The proposed numerical schemes are shown to preserve two invariants, momentum and energy, hence numerically producing wave solutions with smaller phase error over a long time period than those generated by other conventional methods. We first apply the scheme to the CH equation and showcase the merits of considering such a scheme under a wide class of initial data. We then generalize this scheme to the 2CH equation and test this scheme under several types of initial data.

We propose a high order discontinuous Galerkin method for solving nonlinear Fokker---Planck equations with a gradient flow structure. For some of these models it is known that the transient solutions converge to steady-states when time tends to infinity. The scheme is shown to satisfy a discrete version of the entropy dissipation law and preserve steady-states, therefore providing numerical solutions with satisfying long-time behavior. The positivity of numerical solutions is enforced through a reconstruction algorithm, based on positive cell averages. For the model with trivial potential, a parameter range sufficient for positivity preservation is rigorously established. For other cases, cell averages can be made positive at each time step by tuning the numerical flux parameters. A selected set of numerical examples is presented to confirm both the high-order accuracy and the efficiency to capture the large-time asymptotic.

In this work, we propose a high order alternating evolution discontinuous Galerkin (AEDG) method to solve convection-diffusion equations. The difficulties related to numerical fluxes of DG methods for diffusion problems have been a major issue of investigation in the literature. The AEDG scheme presented here is based on an alternating evolution system for convection-diffusion equations, and therefore no numerical fluxes are needed in the scheme formulation. Moreover, the method is shown to be consistent, conservative and stable. Numerical experiments are provided to show the goodness of the proposed method.

We propose and study a nonlocal Euler system with relaxation, which tends to a strictly hyperbolic system under the hyperbolic scaling limit. An independent proof of the local existence and uniqueness of this system is presented in any spatial dimension. We further derive a precise critical threshold for this system in one dimensional setting. Our result reveals that such nonlocal system admits global smooth solutions for a large class of initial data. Thus, the nonlocal velocity regularizes the generic finite-time breakdown in the pressureless Euler system.

The state space for solutions of the compressible Euler equations with a general equation of state is examined. An arbitrary equation of state is allowed, subject only to the physical requirements of thermodynamics. An invariant region of the resulting Euler system is identified and the convexity property of this region is justified by using only very minimal thermodynamical assumptions. Finally, we show how an invariant-region-preserving (IRP) limiter can be constructed for use in high order finite-volume type schemes to solve the compressible Euler equations with a general constitutive relation.

In this paper, we design and analyze second order positive and free energy satisfying schemes for solving diffusion equations with interaction potentials. The semi-discrete scheme is shown to conserve mass, preserve solution positivity, and satisfy a discrete free energy dissipation law for nonuniform meshes. These properties for the fully-discrete scheme (first order in time) remain preserved without a strict restriction on time steps. For the fully second order (in both time and space) scheme, we use a local scaling limiter to restore solution positivity when necessary. It is proved that such limiter does not destroy the second order accuracy. In addition, these schemes are easy to implement, and efficient in simulations over long time. Both one and two dimensional numerical examples are presented to demonstrate the performance of these schemes., Comment: 29 pages, 3 tables, 6 figures

In this work, we design, analyze, and numerically test an invariant preserving discontinuous Galerkin method for solving the nonlinear Camassa--Holm equation. This model is integrable and admits peakon solitons. The proposed numerical method is high order accurate, and preserves two invariants, momentum and energy, of this nonlinear equation. The $L^2$-stability of the scheme for general solutions is a consequence of the energy preserving property. The numerical simulation results for different types of solutions of the Camassa--Holm equation are provided to illustrate the accuracy and capability of the proposed method.

We identify a sub-threshold for finite time shock formation in solutions to the one-dimensional hyperbolic Keller–Segel model. The main result states that under some assumptions on the initial potential, if the slope of the initial density is above a threshold at even one location, the solution must become discontinuous in finite time.

This paper is concerned with the discrete dynamics of an integro-differential model that describes the evolution of a population structured with respect to a continuous trait. Various time-asymptotic convergence rates towards the discrete evolutionary stable distribution (ESD) are established. For some special ESD satisfying a strict sign condition, the exponential convergence rates are obtained for both semi-discrete and fully discrete schemes. Towards the general ESD, the algebraic convergence rate that we find is consistent with the known result for the continuous model.

In this paper, we present the optimal L2-error estimate ofO(hk+1) for polynomial elements of degree k of the semidiscrete direct discontinuous Galerkin method for convection-diffusion equations. The main technical difficulty lies in the control of the inter-element jump terms which arise because of the convection and the discontinuous nature of numerical solutions. The main idea is to use some global projections satisfying interface conditions dictated by the choice of numerical fluxes so that trouble terms at the cell interfaces are eliminated or controlled. In multi-dimensional case, the orders of k + 1 hinge on a superconvergence estimate when tensor product polynomials of degree k are used on Cartesian grids. A collection of projection errors in both oneand multi-dimensional cases is established.

We construct, analyze and numerically validate a class of mass preserving, direct discontinuous Galerkin (DDG) schemes for Schrodinger equations subject to both linear and nonlinear potentials. Up to round-off error, these schemes preserve the discrete version of the mass of the continuous solution. For time discretization, we use the Crank-Nicolson for linear Schrodinger equations, and the Strang splitting for nonlinear Schrodinger equations, so that numerical mass is still preserved at each time step. The DDG method when applied to linear Schrodinger equations is shown to have the optimal ( k + 1 )th order of accuracy for polynomial elements of degree k. The numerical tests demonstrate both accuracy and capacity of these methods.

An invariant-region-preserving (IRP) limiter for multi-dimensional hyperbolic conservation law systems is introduced, as long as the system admits a global invariant region which is a convex set in the phase space. It is shown that the order of approximation accuracy is not destroyed by the IRP limiter, provided the cell average is away from the boundary of the convex set. Moreover, this limiter is explicit, and easy for computer implementation. A generic algorithm incorporating the IRP limiter is presented for high order finite volume type schemes. For arbitrarily high order discontinuous Galerkin (DG) schemes to hyperbolic conservation law systems, sufficient conditions are obtained for cell averages to remain in the invariant region provided the projected one-dimensional system shares the same invariant region as the full multi-dimensional hyperbolic system {does}. The general results are then applied to both one and two dimensional compressible Euler equations so to obtain high order IRP DG schemes. Numerical experiments are provided to validate the proven properties of the IRP limiter and the performance of IRP DG schemes for compressible Euler equations., Comment: 33 pages, 8 tables, 5 figures, accepted for publication in Journal of Computational Physics, 2018

In this paper, we present entropy satisfying schemes for solving an integro-differential equation that describes the evolution of a population structured with respect to a continuous trait. In [P.-E. Jabin and G. Raoul, J. Math. Biol., 63 (2011), pp. 493--517] solutions are shown to converge toward the so-called evolutionary stable distribution (ESD) as time becomes large, using the relative entropy. At the discrete level, the ESD is shown to be the solution to a quadratic programming problem and can be computed by any well-established nonlinear programing algorithm. The schemes are then shown to satisfy the entropy dissipation inequality on the set where initial data are positive and the numerical solutions tend toward the discrete ESD in time. An alternative algorithm is presented to capture the global ESD for nonnegative initial data, which is made possible due to the mutation mechanism built into the modified scheme. A series of numerical tests are given to confirm both accuracy and the entropy satisf...

Traveling wave solutions to a class of dispersive models, u t − u t x x + u u x = θ u u x x x + ( 1 − θ ) u x u x x , are investigated in terms of the parameter θ, including two integrable equations, the Camassa–Holm equation, θ = 1 / 3 , and the Degasperis–Procesi equation, θ = 1 / 4 , as special models. It was proved in H. Liu and Z. Yin (2011) [39] that when 1 / 2 θ ≤ 1 smooth solutions persist for all time, and when 0 ≤ θ ≤ 1 2 , strong solutions of the θ-equation may blow up in finite time, yielding rich traveling wave patterns. This work therefore restricts to only the range θ ∈ [ 0 , 1 / 2 ] . It is shown that when θ = 0 , only periodic travel wave is permissible, and when θ = 1 / 2 traveling waves may be solitary, periodic or kink-like waves. For 0 θ 1 / 2 , traveling waves such as periodic, solitary, peakon, peaked periodic, cusped periodic, or cusped soliton are all permissible.

We design, analyze and numerically validate a novel discontinuous Galerkin method for solving the coagulation-fragmentation equations. The DG discretization is applied to the conservative form of the model, with flux terms evaluated by Gaussian quadrature with $Q=k+1$ quadrature points for polynomials of degree $k$. The positivity of the numerical solution is enforced through a simple scaling limiter based on positive cell averages. The positivity of cell averages is propagated by the time discretization provided a proper time step restriction is imposed., 16 pages, 2 figures, 6 tables

In this work we design and analyze a free energy satisfying finite difference method for solving Poisson–Nernst–Planck equations in a bounded domain. The algorithm is of second order in space, with numerical solutions satisfying all three desired properties: i) mass conservation, ii) positivity preserving, and iii) free energy satisfying in the sense that these schemes satisfy a discrete free energy dissipation inequality. These ensure that the computed solution is a probability density, and the schemes are energy stable and preserve the equilibrium solutions. Both one- and two-dimensional numerical results are provided to demonstrate the good qualities of the algorithm, as well as effects of relative size of the data given.

In Liu and Yu (SIAM J Numer Anal 50(3):1207---1239, 2012), we developed a finite volume method for Fokker---Planck equations with an application to finitely extensible nonlinear elastic dumbbell model for polymers subject to homogeneous fluids. The method preserves positivity and satisfies the discrete entropy inequalities, but has only first order accuracy in general cases. In this paper, we overcome this problem by developing uniformly accurate, entropy satisfying discontinuous Galerkin methods for solving Fokker---Planck equations. Both semidiscrete and fully discrete methods satisfy two desired properties: mass conservation and entropy satisfying in the sense that these schemes are shown to satisfy the discrete entropy inequality. These ensure that the schemes are entropy satisfying and preserve the equilibrium solutions. It is also proved the convergence of numerical solutions to the equilibrium solution as time becomes large. At the finite time, a positive truncation is used to generate the nonnegative numerical approximation which is as accurate as the obtained numerical solution. Both one and two-dimensional numerical results are provided to demonstrate the good qualities of the schemes, as well as effects of some canonical homogeneous flows.

In this work, we design, analyze and test a local discontinuous Galerkin method for solving the Burgers---Poisson equation. This model, proposed by Whitham [Linear and nonlinear waves, 1974] as a simplified model for shallow water waves, admits conservation of both momentum and energy as two invariants. The proposed numerical method is high order accurate and preserves two invariants, hence producing solutions with satisfying long time behavior. The $$L^2$$ L 2 -stability of the scheme for general solutions is a consequence of the energy preserving property. The optimal order of accuracy for polynomial elements of even degree is proven. A series of numerical tests is provided to illustrate both accuracy and capability of the method.