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In this paper, we propose a novel discontinuous Galerkin (DG) method to control the spurious oscillations when solving the scalar hyperbolic conservation laws. Usually, the high order linear numeri...

In this paper we consider the Runge--Kutta discontinuous Galerkin (RKDG) method to solve linear constant-coefficient hyperbolic equations, where the fourth-order explicit Runge--Kutta time-marching...

This paper is devoted to the construction and analysis of new nonlinear optimal weights for weighted ENO (WENO) interpolation capable of raising the order of accuracy close to discontinuities. The new nonlinear optimal weights are constructed using a strategy inspired by the original WENO algorithm, and they work very well for corner or jump singularities, leading to optimal theoretical accuracy. This is the first part of a series of two papers. In this first part we analyze the performance of the new algorithms proposed for univariate function approximation in the point values (interpolation problem). In the second part, we will extend the analysis to univariate function approximation in the cell averages (reconstruction problem). Our aim is twofold: to raise the order of accuracy of the WENO type interpolation schemes both near discontinuities and in the interval which contains the singularity. The first problem can be solved using the new nonlinear optimal weights, but the second one requires a new strategy that locates the position of the singularity inside the cell in order to attain adaption. This new strategy is inspired by the ENO-SR schemes proposed by Harten [J. Comput. Phys., 83 (1989), pp. 148--184]. Thus, we will introduce two different algorithms in the point values. The first one can deal with corner singularities and jump discontinuities for intervals not containing the singularity. The second algorithm can also deal with intervals containing corner singularities, as they can be detected from the point values, but jump discontinuities cannot, as the information of their position is lost during the discretization process. As mentioned before, the second part of this work will be devoted to the cell averages and, in this context, it will be possible to work with jump discontinuities as well. The work of the authors was supported by the Programa de Apoyo a la Investigatión de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18, by the national research project MTM2015-64382-P (MINECO/FEDER), and by National Science Foundation grant DMS-1719410.

Motivated by studies on fully discrete numerical schemes for linear hyperbolic conservation laws, we present a framework on analyzing the strong stability of explicit Runge-Kutta (RK) time discretizations for semi-negative autonomous linear systems. The analysis is based on the energy method and can be performed with the aid of a computer. Strong stability of various RK methods, including a sixteen-stage embedded pair of order nine and eight, has been examined under this framework. Based on numerous numerical observations, we further characterize the features of strongly stable schemes. A both necessary and sufficient condition is given for the strong stability of RK methods of odd linear order.

In this paper, we study the superconvergence behavior of the semi-discrete discontinuous Galerkin (DG) method for scalar nonlinear hyperbolic equations in one spatial dimension. Superconvergence results for problems with fixed and alternating wind directions are established. On the one hand, we prove that, if the wind direction is fixed (i.e., the derivative of the flux function is bounded away from zero), both the cell average error and numerical flux error at cell interfaces converge at a rate of $2k+1$ when upwind fluxes and piecewise polynomials of degree $k$ are used. Moreover, we also prove that the function value approximation of the DG solution is superconvergent at interior right Radau points, and the derivative value approximation is superconvergent at interior left Radau points, with an order of k+2 and k+1, respectively. As a byproduct, we show a (k+2)th order superconvergence of the DG solution towards the Gauss--Radau projection of the exact solution. On the other hand, superconvergence resu...

We analyze the central discontinuous Galerkin method for time-dependent linear conservation laws. In one dimension, optimal a priori $L^2$ error estimates of order $k+1$ are obtained for the semidiscrete scheme when piecewise polynomials of degree at most $k$ ($k\geq0$) are used on overlapping uniform meshes. We then extend the analysis to multidimensions on uniform Cartesian meshes when piecewise tensor-product polynomials are used on overlapping meshes. Numerical experiments are given to demonstrate the theoretical results.

The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with carefully chosen implicit-explicit (IMEX) Runge--Kutta time discretization up to third order accuracy for solving one-dimensional linear advection-diffusion equations. In the time discretization the advection term is treated explicitly and the diffusion term implicitly. There are three highlights of this work. The first is that we establish an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG methods. The second is that, by aid of the aforementioned relationship and the energy method, we show that the IMEX LDG schemes are unconditionally stable for the linear problems in the sense that the time-step $\tau$ is only required to be upper-bounded by a constant which depends on the ratio of the diffusion and the square of the advection coefficients and is independent...

In this paper, we introduce a general approach for proving optimal $L^2$ error estimates for the semidiscrete local discontinuous Galerkin (LDG) methods solving linear high order wave equations. The optimal order of error estimates holds not only for the solution itself but also for the auxiliary variables in the LDG method approximating the various order derivatives of the solution. Examples including the one-dimensional third order wave equation, one-dimensional fifth order wave equation, and multidimensional Schrodinger equation are explored to demonstrate this approach. The main idea is to derive energy stability for the various auxiliary variables in the LDG discretization by using the scheme and its time derivatives with different test functions. Special projections are utilized to eliminate the jump terms at the cell boundaries in the error estimate in order to achieve the optimal order of accuracy.

In this paper, we study the superconvergence of the error for the discontinuous Galerkin (DG) finite element method for linear conservation laws when upwind fluxes are used. We prove that if we apply piecewise $k$th degree polynomials, the error between the DG solution and the exact solution is ($k+2$)th order superconvergent at the downwind-biased Radau points with suitable initial discretization. Moreover, we also prove the DG solution is ($k+2$)th order superconvergent both for the cell averages and for the error to a particular projection of the exact solution. The superconvergence result in this paper leads to a new a posteriori error estimate. Our analysis is valid for arbitrary regular meshes and for $\mathcal{P}^k$ polynomials with arbitrary $k\geq1$, and for both periodic boundary conditions and for initial-boundary value problems. We perform numerical experiments to demonstrate that the superconvergence rate proved in this paper is optimal.

In this paper, an analysis of the superconvergence property of the semidiscrete discontinuous Galerkin (DG) method applied to one-dimensional time-dependent nonlinear scalar conservation laws is carried out. We prove that the error between the DG solution and a particular projection of the exact solution achieves $\left(k + \frac32\right)$th order superconvergence when upwind fluxes are used. The results hold true for arbitrary nonuniform regular meshes and for piecewise polynomials of degree $k$ ($k \ge 1$), under the condition that $|f'(u)|$ possesses a uniform positive lower bound. Numerical experiments are provided to show that the superconvergence property actually holds true for nonlinear conservation laws with general flux functions, indicating that the restriction on $f(u)$ is artificial.

In this paper, we study the superconvergence property for the discontinuous Galerkin (DG) and the local discontinuous Galerkin (LDG) methods for solving one-dimensional time dependent linear conservation laws and convection-diffusion equations. We prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection-diffusion equations. The order of superconvergence for both cases is proved to be $k+\frac{3}{2}$ when piecewise $P^k$ polynomials with $k\geq1$ are used. The proof is valid for arbitrary nonuniform regular meshes and for piecewise $P^k$ polynomials with arbitrary $k\geq1$, improving upon the results in [Y. Cheng and C.-W. Shu, J. Comput. Phys., 227 (2008), pp. 9612-9627], [Y. Cheng and C.-W. Shu, Computers and Structures, 87 (2009), pp. 630-641] in which the proof based on Fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise $P^1$ polynomials.

In this paper we present an analysis of the Runge-Kutta discontinuous Galerkin method for solving scalar conservation laws, where the time discretization is the third order explicit total variation diminishing Runge-Kutta method. We use an energy technique to prove the $\mathrm{L}^2$-norm stability for scalar linear conservation laws and to obtain a priori error estimates for smooth solutions of scalar nonlinear conservation laws. Quasi-optimal order is obtained for general numerical fluxes, and optimal order is given for upwind fluxes. The theoretical results are obtained for piecewise polynomials with any degree $k\geq1$ under the standard temporal-spatial CFL condition $\tau\leq\gamma h$, where $h$ and $\tau$ are the element length and time step, respectively, and the positive constant $\gamma$ is independent of $h$ and $\tau$.

It is well known that finite difference or finite volume total variation diminishing (TVD) schemes solving one-dimensional scalar conservation laws degenerate to first order accuracy at smooth extrema [S. Osher and S. Chakravarthy, SIAM J. Numer. Anal., 21 (1984), pp. 955-984], thus TVD schemes are at most second order accurate in the $L^1$ norm for general smooth and nonmonotone solutions. However, Sanders [Math. Comp., 51 (1988), pp. 535-558] introduced a third order accurate finite volume scheme which is TVD, where the total variation is defined by measuring the variation of the reconstructed polynomials rather than the traditional way of measuring the variation of the grid values. By adopting the definition of the total variation for the numerical solutions as in [R. Sanders, Math. Comp., 51 (1988), pp. 535-558], it is possible to design genuinely high order accurate TVD schemes. In this paper, we construct a finite volume scheme which is TVD in this sense with high order accuracy (up to sixth order) in the $L^1$ norm. Numerical tests for a fifth order accurate TVD scheme will be reported, which include test cases from traffic flow models.

In this paper, we develop, analyze, and test a local discontinuous Galerkin (LDG) method for solving the Camassa–Holm equation which contains nonlinear high-order derivatives. The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the $L^2$ stability for general solutions and give a detailed error estimate for smooth solutions, and provide numerical simulation results for different types of solutions of the nonlinear Camassa–Holm equation to illustrate the accuracy and capability of the LDG method.

The central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408-463] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of Kurganov and Tadmor [J. Comput. Phys., 160 (2000), pp. 241-282] employs a variable control volume, which in turn yields a semidiscrete nonstaggered central scheme. Another approach, which we advocate here, is to view the staggered meshes as a collection of overlapping cells and to realize the computed solution by its overlapping cell averages. This leads to a simple technique to avoid the excessive numerical dissipation for small time steps [Y. Liu, J. Comput. Phys., 209 (2005), pp. 82-104]. At the heart of the proposed approach is the evolution of two pieces of information per cell, instead of one cell average which characterizes all central and upwind Godunov-type finite volume schemes. Overlapping cells lend themselves to the development of a central-type discontinuous Galerkin (DG) method, following the series of works by Cockburn and Shu [J. Comput. Phys., 141 (1998), pp. 199-224] and the references therein. In this paper we develop a central DG technique for hyperbolic conservation laws, where we take advantage of the redundant representation of the solution on overlapping cells. The use of redundant overlapping cells opens new possibilities beyond those of Godunov-type schemes. In particular, the central DG is coupled with a novel reconstruction procedure which removes spurious oscillations in the presence of shocks. This reconstruction is motivated by the moments limiter of Biswas, Devine, and Flaherty [Appl. Numer. Math., 14 (1994), pp. 255-283] but is otherwise different in its hierarchical approach. The new hierarchical reconstruction involves a MUSCL or a second order ENO reconstruction in each stage of a multilayer reconstruction process without characteristic decomposition. It is compact, easy to implement over arbitrary meshes, and retains the overall preprocessed order of accuracy while effectively removing spurious oscillations around shocks.

In this paper we discuss two explicit finite difference schemes, namely a first order upwind scheme and a second order high resolution scheme, for solving a hierarchical size-structured population model with nonlinear growth, mortality, and reproduction rates. We prove stability and convergence for both schemes and provide numerical examples to demonstrate their capability in solving smooth and discontinuous solutions.

In this paper we study the error estimates to sufficiently smooth solutions of symmetrizable systems of conservation laws for the Runge–Kutta discontinuous Galerkin (RKDG) method. Time discretization is the second‐order explicit TVD (total variation diminishing) Runge–Kutta method, and the $\mathbb{P}^k$ (piecewise polynomial) finite element is used. When $k=1$ (piecewise linear finite element), the error estimate is obtained under the usual CFL condition $\dt\leq \beta h$ for nonlinear systems in one dimension and for linear systems in multiple space dimensions. Here, h is the maximum element length, τ is the time step, and β is a positive constant independent of h and τ. Error estimates for $\mathbb{P}^k$ finite elements with $k>1$ are obtained under a more restrictive CFL condition.

In this paper we study the error estimates to sufficiently smooth solutions of scalar conservation laws for Runge--Kutta discontinuous Galerkin (RKDG) methods, where the time discretization is the second order explicit total variation diminishing (TVD) Runge--Kutta method. Error estimates for the $\mathbb{P}^1$ (piecewise linear) elements are obtained under the usual CFL condition $\tau\leq \gamma h$ for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions, where h and $\tau$ are the maximum element lengths and time steps, respectively, and the positive constant $\gamma$ is independent of $h$ and $\tau$. However, error estimates for higher order $\mathbb{P}^k(k\geq 2)$ elements need a more restrictive time step $\tau\leq \gamma h^{4/3}$. We remark that this stronger condition is indeed necessary, as the method is linearly unstable under the usual CFL condition $\tau\leq\gamma h$ for the $\mathbb{P}^k$ elements of degree $k\geq 2$. Error estimates of $O(h^{k+1/2}+\tau^2)$ are obtained for general monotone numerical fluxes, and optimal error estimates of $O(h^{k+1}+\tau^2)$ are obtained for upwind numerical fluxes.

In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge--Kutta discontinuous Galerkin (RKDG) methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, high-order formal accuracy, and easy handling of complicated geometries for convection-dominated problems. It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.

We continue the investigation of overcoming the Gibbs phenomenon, i.e., obtaining exponential accuracy at all points, including at the discontinuities themselves, from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. We show that if we are given the first N expansion coefficients of an $L^2 $ function $f(x)$ in terms of either the trigonometric polynomials or the Legendre polynomials, we can construct an exponentially convergent approximation to the point values of $f(x)$ in any sub-interval in which it is analytic.

The applications of high-order, compact finite difference methods in shock calculations are discussed. The main concern is to define a local mean which will serve as a reference for introducing a local nonlinear limiting to control spurious numerical oscillations while maintaining the formal accuracy of the scheme. For scalar conservation laws, the resulting schemes can be proven total-variation stable in one space dimension and maximum-norm stable in multiple space dimensions. Numerical examples are shown to verify accuracy and stability of such schemes for problems containing shocks. These ideas can also be applied to other implicit schemes such as the continuous Galerkin finite element methods.

The existence of one-sided filters, for spectral Fourier approximations of discontinuous functions, which can recover spectral accuracy up to discontinuity from one side, was proved. A least square procedure was also used to construct such a filter and test it on several discontinuous functions numerically.

Hamilton–Jacobi (H–J) equations are frequently encountered in applications, e.g., in control theory and differential games. H–J equations are closely related to hyperbolic conservation laws—in one space dimension the former is simply the integrated version of the latter. Similarity also exists for the multidimensional case, and this is helpful in the design of difference approximations. In this paper high-order essentially nonoscillatory (ENO) schemes for H–J equations are investigated, which yield uniform high-order accuracy in smooth regions and sharply resolve discontinuities in the derivatives. The ENO scheme construction procedure is adapted from that for hyperbolic conservation laws. The schemes are numerically tested on a variety of one-dimensional and two-dimensional problems, including a problem related to control optimization, and high-order accuracy in smooth regions, good resolution of discontinuities in the derivatives, and convergence to viscosity solutions are observed.