The goal of this work is to introduce new families of shock-capturing high-order numerical methods for systems of conservation laws that combine Fast WENO (FWENO) and Optimal WENO (OWENO) reconstructions with Approximate Taylor methods for the time discretization. FWENO reconstructions are based on smoothness indicators that require a lower number of calculations than the standard ones. OWENO reconstructions are based on a definition of the nonlinear weights that allows one to unconditionally attain the optimal order of accuracy regardless of the order of critical points. Approximate Taylor methods update the numerical solutions by using a Taylor expansion in time in which, instead of using the Cauchy-Kovalevskaya procedure, the time derivatives are computed by combining spatial and temporal numerical differentiation with Taylor expansions in a recursive way. These new methods are compared between them and against methods based on standard WENO implementations and/or SSP-RK time discretization. A number of test cases are considered ranging from scalar linear 1d problems to nonlinear systems of conservation laws in 2d.
We present a (partial) historical summary of the mathematical analysis of finite differences and finite volumes methods, paying a special attention to the Lax-Richtmyer and Lax-Wendroff theorems. We then state a Lax-Wendroff consistency result for convection operators on staggered grids (often used in fluid flow simulations), which illustrates a recent generalization of the flux consistency notion designed to cope with general discrete functions.
We prove a sharp uniform generalized Gaussian bound for the Green's function of the Lax-Wendroff and Beam-Warming schemes. Our bound highlights the spatial region that leads to the well-known (rather weak) instability of these schemes in the maximum norm. We also recover uniform bounds in the maximum norm when these schemes are applied to initial data of bounded variation.
We study the stability of the two-dimensional Lax-Wendroff scheme with a stabilizer that approximates solutions to the transport equation. The problem is first analyzed in the whole space in order to show that the so-called energy method yields an optimal stability criterion for this finite difference scheme. We then deal with the case of a half-space when the transport operator is outgoing. At the numerical level, we enforce the Neumann extrapolation boundary condition and show that the corresponding scheme is stable. Eventually we analyze the case of a quarter-space when the transport operat…
We study the water hammer problem in the case of a sudden closing of a valve upstream, and we consider a Lax-Wendroff finite difference scheme in order to obtain a numerical solution of this problem. In order to establish the approximation of this scheme to the original case, we rigorously show some properties such as consistency, stability and weak convergence of the scheme under reasonable conditions. In addition, we present some numerical simulations in order to show some features of the numerical method.
The Lax-Wendroff method is a single step method for evolving time dependent solutions governed by partial differential equations, in contrast to Runge- Kutta methods that need multiple stages per time step. We develop a flux reconstruction version of the method in combination with a Jacobian-free Lax- Wendroff procedure that is applicable to general hyperbolic conservation laws. The method is of collocation type, is quadrature free and can be cast in terms of matrix and vector operations. Special attention is paid to the construction of numerical flux, including for non-linear problems, result…
A combined Lax-Wendroff/interpolation approach with finite element method for a three-dimensional system of tectonic deformation model: application to landslides in Cameroon
A Two-Stage Fourth Order Time-Accurate Discretization for Lax-Wendroff Type Flow Solvers. I. Hyperbolic Conservation Laws
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics（计算物理实验室，应用物理与计算数学研究所）；School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）
An L-Stable Implicit Two-Stage Fourth-Order Temporal Discretization Scheme for Lax-Wendroff-Type Solvers Applied to Stiff Problems
School of Mathematics and Information Science, Henan Polytechnic University（河南理工大学数学与信息科学学院）；School of Mechatronical Engineering, Beijing Institute of Technology（北京理工大学机械与车辆学院）
Admissible Lax-Wendroff Flux Reconstruction Method with Automatic Differentiation on Adaptive Curved Meshes for Relativistic Hydrodynamics
Indian Institute of Technology Delhi（印度理工学院德里分校）；Johannes Gutenberg University Mainz（美因茨约翰内斯古腾堡大学）；IITD-Abu Dhabi Campus（德里印度理工学院阿布扎比分校）；Tata Institute of Fundamental Research（塔塔基础研究所）
Weighted Inverse Lax-Wendroff Boundary Treatment of Discontinuous Galerkin Methods for Conservation Laws
University of Science and Technology of China（中国科学技术大学）；Laoshan Laboratory（崂山实验室）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
The global spectral analysis (GSA) of numerical methods ensures that the dispersion relation preserving (DRP) property is calibrated in addition to ensuring numerical stability, as advocated in the von Neumann analysis. The DRP nature plays a major role where spatio-temporal dependence in the governing equation and boundary conditions has to be retained, such as in direct numerical simulations (DNS) and large eddy simulations (LES) of fluid flow transition. Using the concept of GSA, methods based on the Lax-Wendroff approach for temporal integration are calibrated using a high accuracy, sixth …
A Positivity-Preserving Limiting Strategy for Locally-Implicit Lax-Wendroff Discontinuous Galerkin Methods
University of Wisconsin–Platteville（威斯康星大学普拉特维尔分校）；Instituto Tecnológico Autónomo de México（墨西哥自治理工学院）；Iowa State University（爱荷华州立大学）；Minnesota State University Moorhead（明尼苏达摩尔黑德州立大学）；University of Florida（佛罗里达大学）
This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We …
The quadrature-based method of moments (QMOM) offers a promising class of approximation techniques for reducing kinetic equations to fluid equations that are valid beyond thermodynamic equilibrium. In this work, we study a particular five-moment variant of QMOM known as HyQMOM and establish that this system is moment-invertible over a convex region in solution space. We then develop a high-order discontinuous Galerkin (DG) scheme for solving the resulting fluid system. The scheme is based on a predictor-corrector approach, where the prediction is a localized space-time DG scheme. The nonlinear…
The analysis of Lax-Wendroff (LW) method is performed by the generic modified differential equation (MDE) approach in the spectral plane using Fourier transform. In this approach, the concept of dispersion relation plays a major role relating spatial and temporal dependence of the governing differential equation, including initial and boundary conditions in developing high accuracy schemes. Such dispersion relation preserving schemes are calibrated in the spectral plane using the global spectral analysis for the numerical method in the full domain. In this framework, the numerical methods are …
Lax convergence theorems and error estimates of a finite element method for the incompressible Euler system
Johannes Gutenberg-University Mainz（美因茨约翰内斯古腾堡大学）；Technical University Darmstadt（达姆施塔特工业大学）；Academy for Multidisciplinary Studies, Capital Normal University（首都师范大学多学科研究院）
The Picard integral formulation of weighted essentially non-oscillatory schemes
Department of Mathematics, Michigan State University（密歇根州立大学数学系）；Department of Electrical and Computer Engineering, Michigan State University（密歇根州立大学电气与计算机工程系）
This paper develops three high-order accurate discontinuous Galerkin (DG) methods for the one-dimensional (1D) and two-dimensional (2D) nonlinear Dirac (NLD) equations with a general scalar self-interaction. They are the Runge-Kutta DG (RKDG) method and the DG methods with the one-stage fourth-order Lax-Wendroff type time discretizaiton (LWDG) and the two-stage fourth-order accurate time discretization (TSDG). The RKDG method uses the spatial DG approximation to discretize the NLD equations and then utilize the explicit multistage high-order Runge-Kutta time discretization for the first-order …
Improved five-point low dissipation nonlinear schemes are proposed in this paper within the framework of weighted compact nonlinear schemes (WCNSs) \cite{Deng2000}. Particularly we follow the work of Li and Du \cite{Li2016} on the two-stage fourth-order temporal accurate discretization scheme, which is developed based on the Lax-Wendroff method.
We present a new family of high-order shock-capturing finite difference numerical methods for systems of conservation laws. These methods, called Adaptive Compact Approximation Taylor (ACAT) schemes, use centered $(2p + 1)$-point stencils, where $p$ may take values in $\{1, 2, \dots, P\}$ according to a new family of smoothness indicators in the stencils. The methods are based on a combination of a robust first order scheme and the Compact Approximate Taylor (CAT) methods of order $2p$-order, $p=1,2,\dots, P$ so that they are first order accurate near discontinuities and have order $2p$ in smo…
In this paper, a least-squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative-norm least-squares, is described in detail. As a consequence, an important numerical conservation theorem is obtained, similar to the famous Lax-Wendroff theorem. The numerical conservation properties of the method in this paper do not …
Compact Approximate Taylor (CAT) methods for systems of conservation laws were introduced by Carrillo and Pares in 2019. These methods, based on a strategy that allows one to extend high-order Lax-Wendroff methods to nonlinear systems without using the Cauchy-Kovalevskaya procedure, have arbitrary even order of accuracy 2p and use (2p + 1)-point stencils, where p is an arbitrary positive integer. More recently in 2021 Carrillo, Macca, Pares, Russo and Zorio introduced a strategy to get rid of the spurious oscillations close to discontinuities produced by CAT methods. This strategy led to the s…
Discontinuous Galerkin schemes for hyperbolic systems in non-conservative variables: quasi-conservative formulation with subcell finite volume corrections
The behavior of physical systems is typically modeled using differential equations which are too complex to solve analytically. In practical problems, these equations are discretized on a computational domain, and numerical solutions are computed. A numerical scheme is called convergent, if in the limit of infinitesimal discretization, the bounds on the discretization error is also infinitesimally small. The approximate solution converges to the "true solution" in this limit. The Lax equivalence theorem enables a proof of convergence given consistency and stability of the method. In this work,…
We present a novel implicit scheme for the numerical solution of time-dependent conservation laws. The core idea of the presented method is to exploit and approximate the mixed spatial-temporal derivative of the solution that occurs naturally when deriving some second order accurate schemes in time. Such an approach is introduced in the context of the Lax-Wendroff (or Cauchy-Kowalevski) procedure when the second time derivative is not completely replaced by space derivatives using the PDE, but the mixed derivative is kept. If approximated in a suitable way, the resulting compact implicit schem…
This paper addresses the three concepts of \textit{ consistency, stability and convergence } in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of "balance laws". Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume approximations employ this viewpoint, and the present paper can be regarded as being in this category. It is first shown that under very mild conditions a weak solution is indeed a solution to the balance law. The schemes considered here al…
We prove in this paper the weak consistency of a general finite volume convection operator acting on discrete functions which are possibly not piecewise-constant over the cells of the mesh and over the time steps. It yields an extension of the Lax-Wendroff if-theorem for general colocated or non-colocated schemes. This result is obtained for general polygonal or polyhedral meshes, under assumptions which, for usual practical cases, essentially boil down to a flux-consistency constraint; this latter is, up to our knowledge, novel and compares the discrete flux at a face to the mean value over t…
Conservation properties of iterative methods applied to implicit finite volume discretizations of nonlinear conservation laws are analyzed. It is shown that any consistent multistep or Runge-Kutta method is globally conservative. Further, it is shown that Newton's method, Krylov subspace methods and pseudo-time iterations are globally conservative while the Jacobi and Gauss-Seidel methods are not in general. If pseudo-time iterations using an explicit Runge-Kutta method are applied to a locally conservative discretization, then the resulting scheme is also locally conservative. However, the co…
A new parametric class of semi-implicit numerical schemes for a level set advection equation on Cartesian grids is derived and analyzed. An accuracy and a stability study is provided for a linear advection equation with a variable velocity using partial Lax-Wendroff procedure and numerical von Neumann stability analysis. The obtained semi-implicit kappa-scheme is 2nd order accurate in space and time in any dimensional case when using a dimension by dimension extension of the one-dimensional scheme that is not the case for analogous fully explicit or fully implicit kappa-schemes. A further impr…
