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We extend the fourth order, two stage Multi-Derivative Runge Kutta (MDRK) scheme to the Flux Reconstruction (FR) framework by writing both stages in terms of a time averaged flux and then using the approximate Lax-Wendroff procedure to compute the time averaged flux. Numerical flux is carefully constructed to enhance Fourier CFL stability and accuracy. A subcell based blending limiter is developed for the MDRK scheme which ensures that the limited scheme is provably admissibility preserving. Along with being admissibility preserving, the blending scheme is constructed to minimize dissipation errors by using Gauss-Legendre solution points and performing MUSCL-Hancock reconstruction on subcells. The accuracy enhancement of the blending scheme is numerically verified on compressible Euler's equations, with test cases involving shocks and small-scale structures.

ADER (Arbitrary high order by DERivatives) and Lax-Wendroff (LW) schemes are two high order single stage methods for solving time dependent partial differential equations. ADER is based on solving a locally implicit equation to obtain a space-time predictor solution while LW is based on an explicit Taylor's expansion in time. We cast the corrector step of ADER Discontinuous Galerkin (DG) scheme into an equivalent quadrature free Flux Reconstruction (FR) framework and then show that the obtained ADER-FR scheme is equivalent to the LWFR scheme with D2 dissipation numerical flux for linear problems. This also implies that the two schemes have the same Fourier stability limit for time step size. The equivalence is verified by numerical experiments.

Lax-Wendroff Flux Reconstruction (LWFR) is a single-stage, high order, quadrature free method for solving hyperbolic conservation laws. This work extends the LWFR scheme to solve conservation laws on curvilinear meshes with adaptive mesh refinement (AMR). The scheme uses a subcell based blending limiter to perform shock capturing and exploits the same subcell structure to obtain admissibility preservation on curvilinear meshes. It is proven that the proposed extension of LWFR scheme to curvilinear grids preserves constant solution (free stream preservation) under the standard metric identities. For curvilinear meshes, linear Fourier stability analysis cannot be used to obtain an optimal CFL number. Thus, an embedded-error based time step computation method is proposed for LWFR method which reduces fine-tuning process required to select a stable CFL number using the wave speed based time step computation. The developments are tested on compressible Euler's equations, validating the blending limiter, admissibility preservation, AMR algorithm, curvilinear meshes and error based time stepping., Comment: 38 pages, 15 figures (with sub-figures), 1 table

This work introduces an extension of the high order, single stage Lax-Wendroff Flux Reconstruction (LWFR) of Babbar et al., JCP (2022) to solve second order time-dependent partial differential equations in conservative form on curvilinear meshes. The method uses BR1 scheme to reduce the system to first order so that the earlier LWFR scheme can be applied. The work makes use of the embedded error-based time stepping introduced in Babbar, Chandrashekar (2024) which becomes particularly relevant in the absence of CFL stability limit for parabolic equations. The scheme is verified to show optimal order convergence and validated with transonic flow over airfoil and unsteady flow over cylinder., Comment: 16 Pages, 8 figures

Lax-Wendroff Flux Reconstruction (LWFR) is a single-stage, high order, quadrature free method for solving hyperbolic conservation laws. We perform a cell average decomposition of the LWFR scheme that is similar to the one used in the admissibility preserving framework of Zhang and Shu (2010). By performing a flux limiting of the time averaged numerical flux, the decomposition is used to obtain an admissibility preserving LWFR scheme. The admissibility preservation framework is further extended to a newly proposed extension of LWFR scheme for conservation laws with source terms. This is the first extension of the high order LW scheme that can handle source terms. The admissibility and accuracy are verified by numerical experiments on the Ten Moment equations of Livermore et al., Comment: 16 pages, 11 figures, accepted for publication in proceedings of ICOSAHOM2023

The shear shallow water model is an extension of the classical shallow water model to include the effects of vertical shear. It is a system of six non-linear hyperbolic PDE with non-conservative products. We develop a high-order entropy stable finite difference scheme for this model in one dimension and extend it to two dimensions on rectangular grids. The key idea is to rewrite the system so that non-conservative terms do not contribute to the entropy evolution. Then, we first develop an entropy conservative scheme for the conservative part, which is then extended to the complete system using the fact that the non-conservative terms do not contribute to the entropy production. The entropy dissipative scheme, which leads to an entropy inequality, is then obtained by carefully adding dissipative flux terms. The proposed schemes are then tested on several one and two-dimensional problems to demonstrate their stability and accuracy., Comment: Accepted in Journal of Scientific Computing (JOMP)

Lax-Wendroff Flux Reconstruction (LWFR) is a single-stage, high order, quadrature free method for solving hyperbolic conservation laws. We develop a subcell based limiter by blending LWFR with a lower order scheme, either first order finite volume or MUSCL-Hancock scheme. While the blending with a lower order scheme helps to control oscillations, it may not guarantee admissibility of discrete solution, e.g., positivity property of quantities like density and pressure. By exploiting the subcell structure and admissibility of lower order schemes, we devise a strategy to ensure that the blended scheme is admissibility preserving for the mean values and then use a scaling limiter to obtain admissibility of the polynomial solution. For MUSCL-Hancock scheme on non-cell-centered subcells, we develop a slope limiter, time step restrictions and suitable blending of higher order fluxes, that ensures admissibility of lower order updates and hence that of the cell averages. By using the MUSCL-Hancock scheme on subcells and Gauss-Legendre points in flux reconstruction, we improve small-scale resolution compared to the subcell-based RKDG blending scheme with first order finite volume method and Gauss-Legendre-Lobatto points. We demonstrate the performance of our scheme on compressible Euler's equations, showcasing its ability to handle shocks and preserve small-scale structures.

Approximate multidimensional Riemann solvers are essential building blocks in designing globally constraint-preserving finite volume time domain (FVTD) and discontinuous Galerkin time domain (DGTD) schemes for computational electrodynamics (CED). In those schemes, we can achieve high-order temporal accuracy with the help of Runge-Kutta or ADER time-stepping. This paper presents the design of a multidimensional approximate Generalized Riemann Problem (GRP) solver for the first time. The multidimensional Riemann solver accepts as its inputs the four states surrounding an edge on a structured mesh, and its output consists of a resolved state and its associated fluxes. In contrast, the multidimensional GRP solver accepts as its inputs the four states and their gradients in all directions; its output consists of the resolved state and its corresponding fluxes and the gradients of the resolved state. The gradients can then be used to extend the solution in time. As a result, we achieve second-order temporal accuracy in a single step. In this work, the formulation is optimized for linear hyperbolic systems with stiff, linear source terms because such a formulation will find maximal use in CED. Our formulation produces an overall constraint-preserving time-stepping strategy based on the GRP that is provably L-stable in the presence of stiff source terms. We present several stringent test problems, showing that the multidimensional GRP solver for CED meets its design accuracy and performs stably with optimal time steps. The test problems include cases with high conductivity, showing that the beneficial L-stability is indeed realized in practical applications., Comment: 38 Pages, 10 figures

In this article, we propose high-order finite-difference entropy stable schemes for the two-fluid relativistic plasma flow equations. This is achieved by exploiting the structure of the equations, which consists of three independent flux components. The first two components describe the ion and electron flows, which are modeled using the relativistic hydrodynamics equation. The third component is Maxwell's equations, which are linear systems. The coupling of the ion and electron flows, and electromagnetic fields is via source terms only. Furthermore, we also show that the source terms do not affect the entropy evolution. To design semi-discrete entropy stable schemes, we extend the RHD entropy stable schemes in Bhoriya et al. to three dimensions. This is then coupled with entropy stable discretization of the Maxwell's equations. Finally, we use SSP-RK schemes to discretize in time. We also propose ARK-IMEX schemes to treat the stiff source terms; the resulting nonlinear set of algebraic equations is local (at each discretization point). These equations are solved using the Newton's Method, which results in an efficient method. The proposed schemes are then tested using various test problems to demonstrate their stability, accuracy and efficiency.

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, resulting in higher CFL numbers than existing methods, which is shown through Fourier analysis and yielding uniform performance at all orders. Numerical results up to fifth order of accuracy for linear and non-linear problems are given to demonstrate the performance and accuracy of the method.

The shear shallow water model is a higher order model for shallow flows which includes some shear effects that are neglected in the classical shallow models. The model is a non-conservative hyperbolic system which can admit shocks, rarefactions, shear and contact waves. The notion of weak solution is based on a path but the choice of the correct path is not known for this problem. In this paper, we construct exact solution for the Riemann problem assuming a linear path in the space of conserved variables, which is also used in approximate Riemann solvers. We compare the exact solutions with those obtained from a path conservative finite volume scheme on some representative test cases.

Split form schemes for Euler and Navier-Stokes equations are useful for computation of turbulent flows due to their better robustness. This is because they satisfy additional conservation properties of the governing equations like kinetic energy preservation leading to a reduction in aliasing errors at high orders. Recently, linear stability issues have been pointed out for these schemes for a density wave problem and we investigate this behaviour for some standard split forms. By deriving linearized equations of split form schemes, we show that most existing schemes do not satisfy a perturbation energy equation that holds at the continuous level. A simple modification to the energy flux of some existing schemes is shown to yield a scheme that is consistent with the energy perturbation equation. Numerical tests are given using a discontinuous Galerkin method to demonstrate these results.

We propose a constraint preserving discontinuous Galerkin method for ideal compressible MHD in two dimensions and using Cartesian grids, which automatically maintains the global divergence-free property. The approximation of the magnetic field is achieved using Raviart-Thomas polynomials and the DG scheme is based on evolving certain moments of these polynomials which automatically guarantees divergence-free property. We also develop HLL-type multi-dimensional Riemann solvers to estimate the electric field at vertices which are consistent with the 1-D Riemann solvers. When limiters are used, the divergence-free property may be lost and it is recovered by a divergence-free reconstruction step. We show the performance of the method on a range of test cases up to fourth order of accuracy.

We introduce novel high order well-balanced finite volume methods for the full compressible Euler system with gravity source term. They require no a priori knowledge of the hydrostatic solution which is to be well-balanced and are not restricted to certain classes of hydrostatic solutions. In one spatial dimension we construct a method that exactly balances a high order discretization of any hydrostatic state. The method is extended to two spatial dimensions using a local high order approximation of a hydrostatic state in each cell. The proposed simple, flexible, and robust methods are not restricted to a specific equation of state. Numerical tests verify that the proposed method improves the capability to accurately resolve small perturbations on hydrostatic states., Comment: In this version we added a discussion of well-balanced boundary conditions (Section 2.3.3) and a new test case (Section 3.5)

The shear shallow water model provides an approximation for shallow water flows by including the effect of vertical shear in the model. This model can be derived from the depth averaging process by including the second order velocity fluctuations which are neglected in the classical shallow water approximation. The resulting model has a non-conservative structure which resembles the 10-moment equations from gas dynamics. This structure facilitates the development of path conservative schemes and we construct HLL, 3-wave and 5-wave HLLC-type solvers. An explicit and semi-implicit MUSCL-Hancock type second order scheme is proposed for the time integration. Several test cases including roll waves show the performance of the proposed modeling and numerical strategy.

We develop a positivity-preserving finite difference WENO scheme for the Ten-Moment equations with body forces acting as a source in the momentum and energy equations. A positive forward Euler scheme under a CFL condition is first constructed which is combined with an operator splitting approach together with an integrating factor, strong stability preserving Runge-Kutta scheme. The positivity of the forward Euler scheme is obtained under a CFL condition by using a scaling type limiter, while the solution of the source operator is performed exactly and is positive without any restriction on the time step. The proposed method can be used with any WENO reconstruction scheme and we demonstrate it with fifth order accurate WENO-JS, WENO-Z and WENO-AO schemes. An adaptive CFL strategy is developed which can be more efficient than the use of reduced CFL for positivity preservation. Numerical results show that high order accuracy and positivity preservation are achieved on a range of test problems.

We propose an explicit, single step discontinuous Galerkin (DG) method on moving grids using the arbitrary Lagrangian-Eulerian (ALE) approach for one dimensional Euler equations. The grid is moved with the local fluid velocity modified by some smoothing, which is found to considerably reduce the numerical dissipation introduced by Riemann solvers. The scheme preserves constant states for any mesh motion and we also study its positivity preservation property. Local grid refinement and coarsening are performed to maintain the mesh quality and avoid the appearance of very small or large cells. Second, higher order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme.

We introduce a general framework for the construction of well-balanced finite volume methods for hyperbolic balance laws. We use the phrase well-balancing in a broader sense, since our proposed method can be applied to exactly follow any solution of any system of hyperbolic balance laws in multiple spatial dimensions and not only time independent solutions. The solution has to be known a priori, either as an analytical expression or as discrete data. The proposed framework modifies the standard finite volume approach such that the well-balancing property is obtained and in case the method is high order accurate, this is maintained under our modification. We present numerical tests for the compressible Euler equations with and without gravity source term and with different equations of state, and for the equations of compressible ideal magnetohydrodynamics.

Computational electrodynamics (CED), the numerical solution of Maxwell's equations, plays an incredibly important role in several problems in science and engineering. High accuracy solutions are desired, and the discontinuous Galerkin (DG) method is one of the better ways of delivering high accuracy in CED. Maxwell's equations have a pair of involution constraints for which mimetic schemes that globally satisfy the constraints at a discrete level are highly desirable. Balsara and Kappeli presented a von Neumann stability analysis of globally constraint-preserving DG schemes for CED up to 4'th order which was focused on developing the theory and documenting the superior dissipation and dispersion of DGTD schemes in media with constant permittivity and permeability. In this paper we present DGTD schemes for CED that go up to 5'th order of accuracy and analyze their performance when permittivity and permeability vary strongly in space. Our DGTD schemes achieve constraint preservation by collocating the electric displacement and magnetic induction as well as their higher order modes in the faces of the mesh. Our first finding is that at 4'th and higher orders, one has to evolve some zone-centered modes in addition to the face-centered modes. It is well-known that the limiting step in DG schemes causes a reduction of the optimal accuracy of the scheme. In this paper we document simulations where permittivity and permeability vary by almost an order of magnitude without requiring any limiting of the DG scheme. This very favorable finding ensures that DGTD schemes retain optimal accuracy even in the presence of large spatial variations in permittivity/permeability. Our third finding shows that the electromagnetic energy is conserved very well even when permittivity and permeability vary strongly in space; as long as the conductivity is zero.

In the present work, we propose two new variants of fifth order finite difference WENO schemes of adaptive order. We compare our proposed schemes with other variants of WENO schemes with special emphasize on WENO-AO(5,3) scheme [Balsara, Garain, and Shu, {\it J. Comput. Phys.}, 326 (2016), pp 780-804]. The first algorithm (WENO-AON(5,3)), involves the construction of a new simple smoothness indicator which reduces the computational cost of WENO-AO(5,3) scheme. Numerical experiments show that accuracy of WENO-AON(5,3) scheme is comparable to that of WENO-AO(5,3) scheme and resolution of solutions involving shock or other discontinuities is comparable to that of WENO-AO(5,3) scheme. The second algorithm denoted as WENO-AO(5,4,3), involves the inclusion of an extra cubic polynomial reconstruction in the base WENO- AO(5,3) scheme, which leads to a more accurate scheme. Extensive numerical experiments in 1D and 2D are performed, which shows that WENO-AO(5,4,3) scheme has better resolution near shocks or discontinuities among the considered WENO schemes with negligible increase in computational cost.

We propose a globally divergence conforming discontinuous Galerkin (DG) method on Cartesian meshes for {\em curl-type hyperbolic conservation} laws based on directly evolving the face and cell moments of the Raviart-Thomas approximation polynomials. The face moments are evolved using a 1-D discontinuous Gakerkin method that uses 1-D and multi-dimensional Riemann solvers while the cell moments are evolved using a standard 2-D DG scheme that uses 1-D Riemann solvers. The scheme can be implemented in a local manner without the need to solve a global mass matrix which makes it a truly DG method and hence useful for explicit time stepping schemes for hyperbolic problems. The scheme is also shown to exactly preserve the divergence of the vector field at the discrete level. Numerical results using second and third order schemes for induction equation are presented to demonstrate the stability, accuracy and divergence preservation property of the scheme.

We develop a second order well-balanced finite volume scheme for compressible Euler equations with a gravitational source term. The well-balanced property holds for arbitrary hydrostatic solutions of the corresponding Euler equations without any restriction on the equation of state. The hydrostatic solution must be known a priori either as an analytical formula or as a discrete solution at the grid points. The scheme can be applied on curvilinear meshes and in combination with any consistent numerical flux function and time stepping routines. These properties are demonstrated on a range of numerical tests., Comment: 32 pages

Modern astrophysical simulations aim to accurately model an ever-growing array of physical processes, including the interaction of fluids with magnetic fields, under increasingly stringent performance and scalability requirements driven by present-day trends in computing architectures. Discontinuous Galerkin methods have recently gained some traction in astrophysics, because of their arbitrarily high order and controllable numerical diffusion, combined with attractive characteristics for high performance computing. In this paper, we describe and test our implementation of a discontinuous Galerkin (DG) scheme for ideal magnetohydrodynamics in the AREPO-DG code. Our DG-MHD scheme relies on a modal expansion of the solution on Legendre polynomials inside the cells of an Eulerian octree-based AMR grid. The divergence-free constraint of the magnetic field is enforced using one out of two distinct cell-centred schemes: either a Powell-type scheme based on nonconservative source terms, or a hyperbolic divergence cleaning method. The Powell scheme relies on a basis of locally divergence-free vector polynomials inside each cell to represent the magnetic field. Limiting prescriptions are implemented to ensure non-oscillatory and positive solutions. We show that the resulting scheme is accurate and robust: it can achieve high-order and low numerical diffusion, as well as accurately capture strong MHD shocks. In addition, we show that our scheme exhibits a number of attractive properties for astrophysical simulations, such as lower advection errors and better Galilean invariance at reduced resolution, together with more accurate capturing of barely resolved flow features. We discuss the prospects of our implementation, and DG methods in general, for scalable astrophysical simulations., Comment: 38 pages, 29 figures, 2 tables, accepted for publication in MNRAS

We present a well-balanced, second order, Godunov-type finite volume scheme for compressible Euler equations with gravity. By construction, the scheme admits a discrete stationary solution which is a second order accurate approximation to the exact stationary solution. Such a scheme is useful for problems involving complex equations of state and/or hydrostatic solutions which are not known in closed form expression. No \'a priori knowledge of the hydrostatic solution is required to achieve the well-balanced property. The performance of the scheme is demonstrated on several test cases in terms of preservation of hydrostatic solution and computation of small perturbations around a hydrostatic solution.

In astrophysics, the two main methods traditionally in use for solving the Euler equations of ideal fluid dynamics are smoothed particle hydrodynamics and finite volume discretization on a stationary mesh. However, the goal to efficiently make use of future exascale machines with their ever higher degree of parallel concurrency motivates the search for more efficient and more accurate techniques for computing hydrodynamics. Discontinuous Galerkin (DG) methods represent a promising class of methods in this regard, as they can be straightforwardly extended to arbitrarily high order while requiring only small stencils. Especially for applications involving comparatively smooth problems, higher-order approaches promise significant gains in computational speed for reaching a desired target accuracy. Here, we introduce our new astrophysical DG code TENET designed for applications in cosmology, and discuss our first results for 3D simulations of subsonic turbulence. We show that our new DG implementation provides accurate results for subsonic turbulence, at considerably reduced computational cost compared with traditional finite volume methods. In particular, we find that DG needs about 1.8 times fewer degrees of freedom to achieve the same accuracy and at the same time is more than 1.5 times faster, confirming its substantial promise for astrophysical applications., Comment: 21 pages, 7 figures, to appear in Proceedings of the SPPEXA symposium, Lecture Notes in Computational Science and Engineering (LNCSE), Springer

We present a well-balanced nodal discontinuous Galerkin (DG) scheme for compressible Euler equations with gravity. The DG scheme makes use of discontinuous Lagrange basis functions supported at Gauss-Lobatto-Legendre (GLL) nodes together with GLL quadrature using the same nodes. The well-balanced property is achieved by a specific form of source term discretization that depends on the nature of the hydrostatic solution, together with the GLL nodes for quadrature of the source term. The scheme is able to preserve isothermal and polytropic stationary solutions upto machine precision on any mesh composed of quadrilateral cells and for any gravitational potential. It is applied on several examples to demonstrate its well-balanced property and the improved resolution of small perturbations around the stationary solution.

Solving the Euler equations of ideal hydrodynamics as accurately and efficiently as possible is a key requirement in many astrophysical simulations. It is therefore important to continuously advance the numerical methods implemented in current astrophysical codes, especially also in light of evolving computer technology, which favours certain computational approaches over others. Here we introduce the new adaptive mesh refinement (AMR) code TENET, which employs a high order discontinuous Galerkin (DG) scheme for hydrodynamics. The Euler equations in this method are solved in a weak formulation with a polynomial basis by means of explicit Runge-Kutta time integration and Gauss-Legendre quadrature. This approach offers significant advantages over commonly employed second order finite volume (FV) solvers. In particular, the higher order capability renders it computationally more efficient, in the sense that the same precision can be obtained at significantly less computational cost. Also, the DG scheme inherently conserves angular momentum in regions where no limiting takes place, and it typically produces much smaller numerical diffusion and advection errors than a FV approach. A further advantage lies in a more natural handling of AMR refinement boundaries, where a fall-back to first order can be avoided. Finally, DG requires no wide stencils at high order, and offers an improved data locality and a focus on local computations, which is favourable for current and upcoming highly parallel supercomputers. We describe the formulation and implementation details of our new code, and demonstrate its performance and accuracy with a set of two- and three-dimensional test problems. The results confirm that DG schemes have a high potential for astrophysical applications., Comment: 23 pages, 12 figures, published in MNRAS November 2015, a movie may be accessed online: https://youtu.be/cTRQP6DSaqA

Centered numerical fluxes can be constructed for compressible Euler equations which preserve kinetic energy in the semi-discrete finite volume scheme. The essential feature is that the momentum flux should be of the form $f^m_\jph = \tp_\jph + \avg{u}_\jph f^\rho_\jph$ where $\avg{u}_\jph = (u_j + u_{j+1})/2$ and $\tp_\jph, f^\rho_\jph$ are {\em any} consistent approximations to the pressure and the mass flux. This scheme thus leaves most terms in the numerical flux unspecified and various authors have used simple averaging. Here we enforce approximate or exact entropy consistency which leads to a unique choice of all the terms in the numerical fluxes. As a consequence novel entropy conservative flux that also preserves kinetic energy for the semi-discrete finite volume scheme has been proposed. These fluxes are centered and some dissipation has to be added if shocks are present or if the mesh is coarse. We construct scalar artificial dissipation terms which are kinetic energy stable and satisfy approximate/exact entropy condition. Secondly, we use entropy- variable based matrix dissipation flux which leads to kinetic energy and entropy stable schemes. These schemes are shown to be free of entropy violating solutions unlike the original Roe scheme. For hypersonic flows a blended scheme is proposed which gives carbuncle free solutions for blunt body flows. Numerical results for Euler and Navier-Stokes equations are presented to demonstrate the performance of the different schemes.

Kinetic schemes for compressible flow of gases are constructed by exploiting the connection between Boltzmann equation and the Navier-Stokes equations. This connection allows us to construct a flux splitting for the Navier-Stokes equations based on the direction of molecular motion from which a numerical flux can be obtained. The naive use of such a numerical flux function in a discontinuous Galerkin (DG) discretization leads to an unstable scheme in the viscous dominated case. Stable schemes are constructed by adding additional terms either in a symmetric or non-symmetric manner which are motivated by the DG schemes for elliptic equations. The novelty of the present scheme is the use of kinetic fluxes to construct the stabilization terms. In the symmetric case, interior penalty terms have to be added for stability and the resulting schemes give optimal convergence rates in numerical experiments. The non-symmetric schemes lead to a cell energy/entropy inequality but exhibit sub-optimal convergence rates. These properties are studied by applying the schemes to a scalar convection-diffusion equation and the 1-D compressible Navier-Stokes equations. In the case of Navier-Stokes equations, entropy variables are used to construct stable schemes.

Vertex-centroid schemes are cell-centered finite volume schemes for conservation laws which make use of vertex values to construct high resolution schemes. The vertex values must be obtained through a consistent averaging (interpolation) procedure. A modified interpolation scheme is proposed which is better than existing schemes in giving positive weights in the interpolation formula. A simplified reconstruction scheme is also proposed which is also more accurate and efficient. For scalar conservation laws, we develop limited versions of the schemes which are stable in maximum norm by constructing suitable limiters. The schemes are applied to compressible flows governed by the Euler equations of inviscid gas dynamics.