Multirate Timestepping Methods for Hyperbolic Conservation Laws

This paper constructs multirate time discretizations for hyperbolic conservation laws that allow different timesteps to be used in different parts of the spatial domain. The proposed family of discretizations is second order accurate in time and has conservation and linear and nonlinear stability properties under local CFL conditions. Multirate timestepping avoids the necessity to take small global timesteps (restricted by the largest value of the Courant number on the grid) and therefore results in more efficient algorithms. Numerical results obtained for the advection and Burgers’ equations confirm the theoretical findings. This work was supported by the National Science Foundation through award NSF CCF-0515170.     

High-Order Multiderivative Time Integrators for Hyperbolic Conservation Laws

Multiderivative time integrators have a long history of development for ordinary differential equations, and yet to date, only a small subset of these methods have been explored as a tool for solving partial differential equations (PDEs). This large class of time integrators include all popular (multistage) Runge–Kutta as well as single-step (multiderivative) Taylor methods. (The latter are commonly referred to as Lax–Wendroff methods when applied to PDEs). In this work, we offer explicit multistage multiderivative time integrators for hyperbolic conservation laws. Like Lax–Wendroff methods, multiderivative integrators permit the evaluation of higher derivatives of the unknown in order to decrease the memory footprint and communication overhead. Like traditional Runge–Kutta methods, multiderivative integrators admit the addition of extra stages, which introduce extra degrees of freedom that can be used to increase the order of accuracy or modify the region of absolute stability. We describe a general framework for how these methods can be applied to two separate spatial discretizations: the discontinuous Galerkin (DG) method and the finite difference essentially non-oscillatory (FD-WENO) method. The two proposed implementations are substantially different: for DG we leverage techniques that are closely related to generalized Riemann solvers; for FD-WENO we construct higher spatial derivatives with central differences. Among multiderivative time integrators, we argue that multistage two-derivative methods have the greatest potential for multidimensional applications, because they only require the flux function and its Jacobian, which is readily available. Numerical results indicate that multiderivative methods are indeed competitive with popular strong stability preserving time integrators.     

Negative-Order Norm Estimates for Nonlinear Hyperbolic Conservation Laws

In this paper, we establish negative-order norm estimates for the accuracy of discontinuous Galerkin (DG) approximations to scalar nonlinear hyperbolic equations with smooth solutions. For these special solutions, we are able to extract this “hidden accuracy” through the use of a convolution kernel that is composed of a linear combination of B-splines. Previous investigations into extracting the superconvergence of DG methods using a convolution kernel have focused on linear hyperbolic equations. However, we now demonstrate that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter for scalar nonlinear hyperbolic equations. Furthermore, we provide theoretical error estimates for the DG solutions that show improvement to $(2k+m)$ -th order in the negative-order norm, where $m$ depends upon the chosen flux.     

Relaxed High Resolution Schemes for Hyperbolic Conservation Laws

Relaxed, essentially non-oscillating schemes for nonlinear conservation laws are presented. Exploiting the relaxation approximation, it is possible to avoid the nonlinear Riemann problem, characteristic decompositions, and staggered grids. Nevertheless, convergence rates up to fourth order are observed numerically. Furthermore, a relaxed, piecewise hyperbolic scheme with artificial compression is constructed. Third order accuracy of this method is proved. Numerical results for two-dimensional Riemann problems in gas dynamics are presented. Finally, the relation to central schemes is discussed.     

Moving Mesh Discontinuous Galerkin Method for Hyperbolic Conservation Laws

In this paper, a moving mesh discontinuous Galerkin (DG) method is developed to solve the nonlinear conservation laws. In the mesh adaptation part, two issues have received much attention. One is about the construction of the monitor function which is used to guide the mesh redistribution. In this study, a heuristic posteriori error estimator is used in constructing the monitor function. The second issue is concerned with the solution interpolation which is used to interpolates the numerical solution from the old mesh to the updated mesh. This is done by using a scheme that mimics the DG method for linear conservation laws. Appropriate limiters are used on seriously distorted meshes generated by the moving mesh approach to suppress the numerical oscillations. Numerical results are provided to show the efficiency of the proposed moving mesh DG method.     

Mesh Redistribution Strategies and Finite Element Schemes for Hyperbolic Conservation Laws

In this work we consider a new class of Relaxation Finite Element schemes for hyperbolic conservation laws, with more stable behavior on the limit area of the relaxation parameter. Combining this scheme with an efficient adapted spatial redistribution process considered also in this work, we form a robust scheme of controllable resolution. The results on a number of test problems show that this scheme can produce entropic-approximations of high resolution, even on the limit of the relaxation parameter where the scheme lacks of the relaxation mechanism. Thus we experimentally conclude that the proposed spatial redistribution process, has by its own interesting stabilization properties for computational solutions of conservation law problems.     

A Speed-Up Strategy for Finite Volume WENO Schemes for Hyperbolic Conservation Laws

In this paper, a speed-up strategy for finite volume WENO schemes is developed for solving hyperbolic conservation laws. It adopts p-adaptive like reconstruction, which automatically adjusts from fifth order WENO reconstruction to first order constant reconstruction when nearly constant solutions are detected by the undivided differences. The corresponding order of accuracy for the solutions is shown to be the same as obtained by original WENO schemes. The strategy is implemented with both WENO and mapped WENO schemes. Numerical examples in different space dimensions show that the strategy can reduce the computational cost by 20–40%, especially for problems with large fraction of constant regions.     

H-adaptive Mesh Method with Double Tolerance Adaptive Strategy for Hyperbolic Conservation Laws

The numerical method used to solve hyperbolic conservation laws is often an explicit scheme. As a commonly used technique to improve the quality of numerical simulation, the $h$ -adaptive mesh method is adopted to resolve sharp structures in the solution. Since the computational costs of altering the mesh and solving the PDEs are comparable, too often the mesh adaption triggered may bring down the overall efficiency of solving hyperbolic conservation laws using $h$ -adaptive mesh method. In this paper, we propose a so-called double tolerance adaptive strategy to optimize the overall numerical efficiency by reducing the number of mesh adaptions, as well as preserving the quality of the numerical solution. Numerical results are presented to demonstrate the robustness and effectiveness of our $h$ -adaptive algorithm using the double tolerance adaptive strategy.     

A P_N P_M{-} CPR Framework for Hyperbolic Conservation Laws

The correction procedure via reconstruction (CPR) method is a discontinuous nodal formulation unifying several well-known methods in a simple finite difference like manner. The \(P_NP_M{-} CPR \) formulation is an extension of \(P_NP_M\) or the reconstructed discontinuous Galerkin (RDG) method to the CPR framework. It is a hybrid finite volume and discontinuous Galerkin (DG) method, in which neighboring cells are used to build a higher order polynomial than the solution representation in the cell under consideration. In this paper, we present several \(P_NP_M\) schemes under the CPR framework. Many interesting schemes with various orders of accuracy and efficiency are developed. The dispersion and dissipation properties of those methods are investigated through a Fourier analysis, which shows that the \(P_NP_M{-} CPR \) method is dependent on the position of the solution points. Optimal solution points for 1D \(P_NP_M{-} CPR \) schemes which can produce expected order of accuracy are identified. In addition, the \(P_NP_M{-} CPR \) method is extended to solve 2D inviscid flow governed by the Euler equations and several numerical tests are performed to assess its performance.     

A Strict Lyapunov Function for Boundary Control of Hyperbolic Systems of Conservation Laws

We present a strict Lyapunov function for hyperbolic systems of conservation laws that can be diagonalized with Riemann invariants. The time derivative of this Lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions. It is shown that the derived boundary control allows to guarantee the local convergence of the state towards a desired set point. Furthermore, the control can be implemented as a feedback of the state only measured at the boundaries. The control design method is illustrated with an hydraulic application, namely the level and flow regulation in an horizontal open channel     

A Stability Criterion for Semi-Discrete Difference Schemes of Hyperbolic Conservation Laws on Uniform Grids

In the present study, the stability condition for semi-discrete difference schemes of hyperbolic conservation laws obtained from Fourier analysis is simplified. This stability condition can be applied only to linear difference schemes with constant coefficients implemented with periodic boundary treatment. It could often give useful results for other cases, such as schemes with variable coefficients, schemes for nonperiodic problem and nonlinear problem. However, this condition usually leads to a trigonometric inequality, which makes it not convenient to use. For explicit difference schemes on uniform grids, this trigonometric inequality can be converted to polynomial form. Furthermore, if the scheme is a high-order one, the polynomial can be factorized into a simple form. Thus, it is much easier to solve than the inequality obtained directly from Fourier analysis. For compact difference schemes and conservative schemes, similar results are obtained. Some applications of this new stability criterion are shown, including judging the stability of two schemes, proving the upstream central schemes to be stable, constructing a stable upwind dissipation relation preserving (DRP) scheme and constructing an optimized weighted essentially non-oscillatory (WENO) scheme. Since WENO schemes are nonlinear schemes, the stability analysis in the present study is performed on their underlying linear schemes. According to the numerical tests, the underlying linear scheme should be stable, otherwise the corresponding WENO scheme may display instability. These applications demonstrate that this criterion is convenient and efficient for judging the linear stability of semi-discrete difference schemes and constructing stable upwind difference schemes.     

变形法移动网格求解双曲型守恒律方程研究

在现有格式的基础上要提高偏微分方程数值解的分辨率,自适应移动网格技术是一种有效而且可行的方法。文中将文献[1]提出的自适应移动网格技术推广到三角形网格,并将该方法用于求解双曲型守恒量方程。用网格自适应技术求解守恒律问题时,当生成新网格之后,需要将旧网格上的函数值更新到新的网格,并保持物理量的守恒性。针对这个问题,文中提出了函数值更新过程中守恒型插值公式的具体形式,并针对二维双曲型守恒律方程进行了仿真实验,取得了满意的结果。     

变形法移动网格求解双曲型守恒律方程研究

在现有格式的基础上要提高偏微分方程数值解的分辨率,自适应移动网格技术是一种有效而且可行的方法。文中将文献[1]提出的自适应移动网格技术推广到三角形网格,并将该方法用于求解双曲型守恒量方程。用网格自适应技术求解守恒律问题时,当生成新网格之后,需要将旧网格上的函数值更新到新的网格,并保持物理量的守恒性。针对这个问题,文中提出了函数值更新过程中守恒型插值公式的具体形式,并针对二维双曲型守恒律方程进行了仿真实验,取得了满意的结果。     

Regularization Techniques for Numerical Approximation of PDEs with Singularities

The rate of convergence for numerical methods approximating differential equations are often drastically reduced from lack of regularity in the solution. Typical examples are problems with singular source terms or discontinuous material coefficients. We shall discuss the technique of local regularization for handling these problems. New numerical methods are presented and analyzed and numerical examples are given. Some serious deficiencies in existing regularization methods are also pointed out.     

A Level Set Algorithm for Tracking Discontinuities in Hyperbolic Conservation Laws II: Systems of Equations

A level set algorithm for tracking discontinuities in hyperbolic conservation laws is presented. The algorithm uses a simple finite difference approach, analogous to the method of lines scheme presented in [36]. The zero of a level set function is used to specify the location of the discontinuity. Since a level set function is used to describe the front location, no extra data structures are needed to keep track of the location of the discontinuity. Also, two solution states are used at all computational nodes, one corresponding to the real state, and one corresponding to a ghost node state, analogous to the Ghost Fluid Method of [12]. High order pointwise convergence was demonstrated for scalar linear and nonlinear conservation laws, even at discontinuities and in multiple dimensions in the first paper of this series [3]. The solutions here are compared to standard high order shock capturing schemes, when appropriate. This paper focuses on the issues involved in tracking discontinuities in systems of conservation laws. Examples will be presented of tracking contacts and hydrodynamic shocks in inert and chemically reacting compressible flow.