Fifth-order Weighted Essentially Non-oscillatory Scheme Research Articles

Application of a Novel Weighted Essentially Non-Oscillatory Scheme for Reynolds-Averaged Stress Model and Reynolds-Averaged Stress Model/Large Eddy Simulation (RANS/LES) Coupled Simulations in Turbomachinery Flows

In numerical simulations, achieving high accuracy without significantly increasing computational cost is often a challenge. To address this issue, this paper proposes an improved finite volume Weighted Essentially Non-Oscillatory (WENO) scheme for structured grids. By employing a single-point quadrature rule to perform flux integration on the control volume faces, this scheme is designed for use in NUAA-Turbo three-dimensional fluid solvers based on structured grids, utilizing RANS and RANS/LES coupling to simulate turbomachinery flows. Firstly, the new WENO scheme is validated against classical numerical test cases to evaluate its stability and reliability in handling discontinuities, double Mach reflection problems, and Rayleigh–Taylor (RT) instability. Compared to the original scheme, this improved finite-volume WENO scheme demonstrates better stability near discontinuities and more effectively resolves flow features at the same grid resolution. Next, for engineering applications related to turbomachinery, such as compressor and turbine characteristics, calculations using RANS are performed and the results obtained with WENO-ZQ3 and WENO-JS3 are compared. Finally, the new fifth-order WENO scheme is applied to RANS/LES coupling simulations of turbine wake and film cooling. The results indicate that the improved finite-volume WENO scheme provides better stability and accuracy in engineering applications. For instance, the average error in calculating compressor efficiency characteristics is reduced from 0.76% to 0.05%, the error in turbine vane pressure distribution compared to the experimental values is within 1%, and the error in film cooling efficiency centerline distribution compared to the experimental values is within 3%. Additionally, the qualitative results of turbine wake and film cooling show that even with a small number of grid points, more detailed flow physics can be captured, thereby reducing computational costs in aerodynamic applications.

Deep smoothness weighted essentially non-oscillatory method for two-dimensional hyperbolic conservation laws: A deep learning approach for learning smoothness indicators

In this work, we enhance the fifth-order Weighted Essentially Non-Oscillatory (WENO) shock-capturing scheme by integrating deep learning techniques. We improve the established WENO algorithm by training a compact neural network to dynamically adjust the smoothness indicators within the WENO scheme. This modification boosts the accuracy of the numerical results, particularly in proximity to abrupt shocks. Notably, our approach eliminates the need for additional post-processing steps, distinguishing it from previous deep learning-based methods. We substantiate the superiority of our new approach through the examination of multiple examples from the literature concerning the two-dimensional Euler equations of gas dynamics. Through a thorough investigation of these test problems, encompassing various shocks and rarefaction waves, our novel technique consistently outperforms the traditional fifth-order WENO scheme. This superiority is especially evident in cases where numerical solutions exhibit excessive diffusion or overshoot around shocks.

A hybrid WENO scheme for steady-state simulations of Euler equations

For strong shock waves in solutions of steady-state Euler equations, the high-order shock capturing schemes usually suffer from the difficulty of convergence of residue close to machine zero. Several new weighted essentially non-oscillatory (WENO) type schemes have recently been designed to overcome this long-standing issue. In this paper, a new hybrid strategy is proposed for the fifth-order WENO scheme to simulate steady-state solutions of Euler equations. Compared with the existing WENO schemes, the hybrid WENO scheme performs better steady-state convergence property with less dissipative and dispersive errors. Meanwhile, the essentially oscillation-free feature is kept. In the hybrid strategy, the stencil is distinguished into smooth, non-smooth, or transition regions, which is realized by a simple smoothness detector based on the smoothness indicators in the original WENO method. The linear reconstruction and the specific WENO reconstruction are applied to the smooth and non-smooth regions, respectively. In the transition region, the mixture of the linear and WENO reconstructions is adopted by a smooth transitive interpolation, which plays a vital role in the steady-state convergence for the hybrid scheme. Numerical comparisons and spectral analysis are presented to demonstrate the robust performance of the new hybrid scheme for steady-state Euler equations.

Numerical investigation on noise reduction of rotor blade-vortex interaction using blade surface jet blowing

The effects of blade surface jet blowing on the reduction of rotor blade-vortex interaction (BVI) noise are investigated using the compressible Reynolds-averaged Navier-Stokes equations and the Ffowcs Williams-Hawkings equations. A detailed analysis of the accuracy of the Weighted Essentially Non-Oscillatory (WENO) scheme and Monotone Upstream-Centered Scheme for Conservation Laws for simulating the flow field and predicting the BVI noise is performed using a two-bladed scaled AH-1 helicopter main rotor undergoing BVI as the baseline case. Additionally, a grid convergence study is conducted. The BVI noise is accurately predicted using the fifth-order WENO scheme and high-resolution grid system (the prediction errors of the peak sound pressure are less than 4%). Surface boundary conditions are introduced to model the effect of jet blowing on the blade surface on the flow control of the baseline case. Blade surface blowing is analyzed near the trailing edge and the leading edge near the rotor tip. The results show that blowing near the blade trailing edge is effective for BVI mitigation by altering the tip vortex strength and increasing the blade-vortex miss distance. The sound pressure level is reduced by 2.6 dB, and the peak amplitude is attenuated by more than 30%, with a 9.8% loss in rotor thrust. Therefore, blowing near the trailing edge may be considered a promising approach for active BVI noise control.

Two-dimensional numerical simulation on gas-solid two-phase flow induced by combustion gas flow in a chamber based on a fifth-order WENO scheme

In order to explore the flow characteristics and propagation law of combustion gas from the central ignition tube in the initial stage of internal ballistic, a visual experimental platform was designed to carry out ignition experiments with the substituted particle bed in the chamber. A high-speed camera system was used to capture the gas flow and flame propagation in the chamber, and a dynamic data acquisition and analysis system with pressure sensors was applied to record the pressure data at characteristic positions in the chamber. A two-dimensional, axisymmetrical, two-phase flow model of internal ballistics was constructed to simulate the flow process of the gas in the substituted particle bed based on a weighted essentially non-oscillatory (WENO) scheme, and the time term was determined by the third-order TVD Runge-Kuta method. The calculated results are in good agreement with the visual experimental results, and the global pressure average error is 5.35%. It indicates that the numerical simulation can accurately describe the gas flow characteristics and present the development process of the gas from the ignition tube in the substituted particle bed. The radial effect of the chamber pressure is obvious, and the gas moves rapidly along the radial direction, and the substituted particle basically does not move in the initial stage of ignition. Moreover, with the gradual propagation of the gas in the chamber, the chamber pressure is characterized by a radial uniformity and an axial gradient distribution. Under the action of the pressure gradient, the axial velocity of the gas phase begins to dominate, and furthermore, the radial velocity decreases to zero in the bottom and the middle region of the chamber, while the solid phase velocity varies with the gas phase velocity. In addition, before the gas reaches the bottom of the right end, the inverse velocity fluctuation appears in advance due to the solid particle congestion.

A fifth-order high-resolution shock-capturing scheme based on modified weighted essentially non-oscillatory method and boundary variation diminishing framework for compressible flows and compressible two-phase flows

First, a new reconstruction strategy is proposed to improve the accuracy of the fifth-order weighted essentially non-oscillatory (WENO) scheme. It has been noted that conventional WENO schemes still suffer from excessive numerical dissipation near-critical regions. One of the reasons is that they tend to under-use all adjacent smooth substencils thus fail to realize optimal interpolation. Hence in this work, a modified WENO (MWENO) strategy is designed to restore the highest possible order interpolation when three target substencils or two target adjacent substencils are smooth. Since the new detector is formulated under the original smoothness indicators, no obvious complexity and cost are added to the simulation. This idea has been successfully implemented into two classical fifth-order WENO schemes, which improve the accuracy near the critical region but without destroying essentially non-oscillatory properties. Second, the tangent of hyperbola for interface capturing (THINC) scheme is introduced as another reconstruction candidate to better represent the discontinuity. Finally, the MWENO and THINC schemes are implemented with the boundary variation diminishing algorithm to further minimize the numerical dissipation across discontinuities. Numerical verifications show that the proposed scheme accurately captures both smooth and discontinuous flow structures simultaneously with high-resolution quality. Meanwhile, the presented scheme effectively reduces numerical dissipation error and suppresses spurious numerical oscillation in the presence of strong shock or discontinuity for compressible flows and compressible two-phase flows.

A new fifth-order alternative finite difference multi-resolution WENO scheme for solving compressible flow

A new fifth-order alternative finite difference multi-resolution weighted essentially non-oscillatory (WENO) scheme is designed to solve the hyperbolic conservation laws in this paper. With the application of the original finite difference multi-resolution WENO schemes (Zhu and Shu, 2018; Zhu and Shu, 2020), a series of unequal-sized central spatial stencils are adopted to perform the WENO procedures, but the difference is that the methodology adopted in this paper is directly based on the point values of the solution rather than on the flux values. The proposed new multi-resolution WENO scheme can inherit many advantages of the original high order schemes, such as the arbitrary choice of the linear weights, the maintenance of the essentially non-oscillatory property in the vicinity of strong discontinuities, the smaller L1 and L∞ truncation errors than that of the same order classical WENOJS schemes (Balsara et al., 2016; Jiang and Shu, 1996) in smooth regions, and better convergence properties of the residue when solving some steady-state problems. Compared with the original multi-resolution WENO schemes, the presented WENO scheme also has some advantages. Firstly, an arbitrary monotone flux can be used in this framework, while the original method is only suitable for smooth flux splitting technique. Secondly, it has smaller L1 and L∞ truncation errors than that of the original same order multi-resolution WENO scheme. And finally, it avoids some time-consuming intermediate processes in the original multi-resolution WENO reconstruction procedures, thus resulting in less calculation time under the same conditions. Some inviscid and viscous numerical examples are provided to verify the superior performance of the new fifth-order alternative finite difference multi-resolution WENO scheme.

Comparison Study of the Fifth-Order WCNS and WENO Scheme In the Truncation Error, Dissipation and Dispersion Terms

This paper makes a comparison study of the fifth-order weighted compact nonlinear scheme (WCNS) and weighted essentially non-oscillatory (WENO) scheme in the truncation error, dissipation and dispersion terms. Truncation error analysis on the WCNS shows that numerical discretization leads to dissipation and dispersion terms. The Fourier spectral analysis methods is used to analyze the dissipative and dispersive features of both schemes.

A fourth-order entropy condition scheme for systems of hyperbolic conservation laws

Based on the analysis of the entropy condition scheme formulation, the accuracy order comes from initial value interpolation and flux reconstruction. Following the limiters of the traditional second-order Total Variation Diminishing scheme, higher accuracy order and non-oscillatory nature are retained with a newly proposed smoothness threshold method. Then, the scheme using the solution formula method in Dong et al. [(2002). High-order discontinuities decomposition entropy condition schemes for Euler equations. CFD Journal, 10(4), 563–568] is generalized to fully-discrete fourth-order accuracy, which retains the advantages of former schemes, i.e. it is a fully-discrete, one-step scheme with no need to perform with a Runge–Kutta process in time; an entropy condition is satisfied with no need of an entropy fix with artificial numerical viscosity; and an exact solution is achieved for linear cases if CFL→1. Numerical experiments are given with a 1D scalar equation for a shock-tube problem, a blast-wave problem, and Shu–Osher problem, a 2D Riemann problem, a double Mach reflection problem and a transonic airfoil flow problem for NACA0012. All tests are compared with a fifth-order Weighted Essentially Non-Oscillatory (WENO) scheme. Numerical experiments and efficiency comparisons show that the efficiency of the new fourth-order scheme is superior to the fifth-order WENO scheme.

A low dissipation finite difference nested multi-resolution WENO scheme for Euler/Navier-Stokes equations

In this paper, a low dissipation fifth-order finite difference nested multi-resolution weighted essentially non-oscillatory (WENO) scheme is presented for solving Euler/Navier-Stokes equations in multi-dimensions on structured meshes. In the reconstruction procedures of this finite difference WENO, a series of nested unequal-sized central stencils are used to design spatial reconstruction polynomials based on the local orthogonal Legendre basis functions. So this new WENO scheme could maintain the original order of accuracy in smooth regions and keep essentially non-oscillatory property by gradually degrading from the fifth-order to third-order, and ultimately to the first-order accuracy near strong discontinuities. Compared with the classical fifth-order finite difference WENO scheme [18], the results show that this new nested multi-resolution WENO scheme could obtain smaller L1 and L∞ errors for solving smooth problems based on the same mesh level. Moreover, the presented WENO scheme also has smaller numerical dissipation without introducing any dissipation preserving technique, and can capture more subtle flow structures for solving viscous flow problems. The proposed multi-resolution WENO scheme can be easily extended to finite volume framework owing to its linear weights can be set as any positive numbers with only one requirement that their summation equals to one. Therefore, the new WENO scheme is more suitable for solving viscous problems containing both discontinuities and complex smooth structures, and can be easily implemented to arbitrarily high-order accuracy in multi-dimensions. Some benchmark inviscid extreme and viscous examples are illustrated to verify the good performance of this nested multi-resolution WENO scheme.

A new fifth-order finite difference well-balanced multi-resolution WENO scheme for solving shallow water equations

In this paper, a new fifth-order finite difference well-balanced multi-resolution weighted essentially non-oscillatory (WENO) scheme is designed to solve for one-dimensional and two-dimensional shallow water equations with or without source terms on structured meshes. We only use the information defined on a hierarchy of nested central spatial stencils without introducing any equivalent multi-resolution representation once again. For the shallow flow problems with smooth or discontinuous bed, we combine with the well-balanced procedure developed by Xing and Shu (2005) for balancing the flux gradients and the source terms, and then the new fifth-order well-balanced multi-resolution WENO scheme (Zhu and Shu, 2018) could satisfy the exact C-property for still stationary solutions, maintain the fifth-order accuracy in smooth regions, and keep essentially non-oscillatory property in non-smooth regions. Compared with the classical well-balanced WENO schemes (Xing and Shu, 2005), the new features of this finite difference well-balanced multi-resolution WENO scheme is its simplicity and hierarchical structure in obtaining higher-order accuracy and its linear weights could be arbitrarily chosen with one requirement that their summation is one. It is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing finite difference well-balanced WENO scheme for solving shallow water equations. This new well-balanced WENO scheme is simple to construct and can be easily implemented to arbitrary high-order accuracy and in higher dimensions. Some benchmark numerical examples are performed to illustrate the good performances of this new well-balanced WENO scheme which could get high-order accuracy, keep exact C-property, sustain good convergence property, and obtain sharp shock transitions in the whole computational field.

A new type of multi-resolution WENO schemes with increasingly higher order of accuracy on triangular meshes

In this paper, we continue our work in [46] and propose a new type of high-order finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes to solve hyperbolic conservation laws on triangular meshes. Although termed “multi-resolution WENO schemes”, we only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. We construct new third-order, fourth-order, and fifth-order WENO schemes using three or four unequal-sized central spatial stencils, different from the classical WENO procedure using equal-sized biased/central spatial stencils for the spatial reconstruction. The new WENO schemes could obtain the optimal order of accuracy in smooth regions, and could degrade gradually to first-order of accuracy so as to suppress spurious oscillations near strong discontinuities. This is the first time that only a series of unequal-sized hierarchical central spatial stencils are used in designing arbitrary high-order finite volume WENO schemes on triangular meshes. The main advantages of these schemes are their compactness, robustness, and their ability to maintain good convergence property for steady-state computation. The linear weights of such WENO schemes can be any positive numbers on the condition that they sum to one. Extensive numerical results are provided to illustrate the good performance of these new finite volume WENO schemes.

Numerical study on the convergence to steady-state solutions of a new class of finite volume WENO schemes: triangular meshes

In this paper, we continue our research on the numerical study of convergence to steady-state solutions for a new class of finite volume weighted essentially non-oscillatory (WENO) schemes in Zhu and Shu (J Comput Phys 349:80–96, 2017), from tensor product meshes to triangular meshes. For the case of triangular meshes, this new class of finite volume WENO schemes was designed for time-dependent conservation laws in Zhu and Qiu (SIAM J Sci Comput 40(2):A903–A928, 2018) for the third- and fourth-order versions. In this paper, we extend the design to a new fifth-order version in the same framework to keep the essentially non-oscillatory property near discontinuities. Similar to the case of tensor product meshes in Zhu and Shu (2017), by performing such spatial reconstruction procedures together with a TVD Runge–Kutta time discretization, these WENO schemes do not suffer from slight post-shock oscillations that are responsible for the phenomenon wherein the residues of classical WENO schemes hang at a truncation error level instead of converging to machine zero. The third-, fourth-, and fifth-order finite volume WENO schemes in this paper can suppress the slight post-shock oscillations and have their residues settling down to a tiny number close to machine zero in steady-state simulations in our extensive numerical experiments.

A sixth-order finite difference WENO scheme for Hamilton–Jacobi equations

ABSTRACTIn this paper, a sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme is developed to approximate the viscosity solution of the Hamilton–Jacobi equations. This new WENO scheme has the same spatial nodes as the classical fifth-order WENO scheme proposed by Jiang and Peng [Weighted ENO schemes for Hamilton–Jacobi equations, SIAM. J. Sci. Comput. 21 (2000), pp. 2126–2143] but can be as high as sixth-order accurate in smooth region while keeping sharp discontinuous transitions with no spurious oscillations near discontinuities. Extensive numerical experiments in one- and two-dimensional cases are carried out to illustrate the capability of the proposed scheme.

A modified fifth-order WENO scheme for hyperbolic conservation laws

Recently a WENO scheme, with smoothness indicators constructed based on L1 measure is introduced by Ha et al. (2013) and the improved version of this scheme is presented by Kim et al. (2016), referred to as WENO-NS and WENO-P schemes respectively. These schemes perform better than the existing many fifth-order WENO schemes for the problems which contain discontinuities and attain fifth-order accuracy at the critical points where the first derivative vanishes but not at the points where the second derivatives are zero. This paper deals with modification of the above said methods to obtain a new fifth-order weighted essentially non-oscillatory (WENO) scheme. A new global-smoothness indicator is proposed which shows an improved behavior over the solutions of WENO-NS and WENO-P schemes and the proposed scheme attains an optimal order of approximation, even at the critical points where the first and second derivatives vanish but not the third derivative. Examples are taken in the numeric section to check the robustness and accuracy of the proposed scheme for one and two-dimensional system of Euler equations.

A simple algorithm to improve the performance of the WENO scheme on non-uniform grids

This paper presents a simple approach for improving the performance of the weighted essentially non-oscillatory (WENO) finite volume scheme on non-uniform grids. This technique relies on the reformulation of the fifth-order WENO-JS (WENO scheme presented by Jiang and Shu in J. Comput. Phys. 126:202–228, 1995) scheme designed on uniform grids in terms of one cell-averaged value and its left and/or right interfacial values of the dependent variable. The effect of grid non-uniformity is taken into consideration by a proper interpolation of the interfacial values. On non-uniform grids, the proposed scheme is much more accurate than the original WENO-JS scheme, which was designed for uniform grids. When the grid is uniform, the resulting scheme reduces to the original WENO-JS scheme. In the meantime, the proposed scheme is computationally much more efficient than the fifth-order WENO scheme designed specifically for the non-uniform grids. A number of numerical test cases are simulated to verify the performance of the present scheme.

High-order weighted essentially nonoscillatory finite-difference formulation of the lattice Boltzmann method in generalized curvilinear coordinates.

In this work, a high-order weighted essentially nonoscillatory (WENO) finite-difference lattice Boltzmann method (WENOLBM) is developed and assessed for an accurate simulation of incompressible flows. To handle curved geometries with nonuniform grids, the incompressible form of the discrete Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) approximation is transformed into the generalized curvilinear coordinates and the spatial derivatives of the resulting lattice Boltzmann equation in the computational plane are solved using the fifth-order WENO scheme. The first-order implicit-explicit Runge-Kutta scheme and also the fourth-order Runge-Kutta explicit time integrating scheme are adopted for the discretization of the temporal term. To examine the accuracy and performance of the present solution procedure based on the WENOLBM developed, different benchmark test cases are simulated as follows: unsteady Taylor-Green vortex, unsteady doubly periodic shear layer flow, steady flow in a two-dimensional (2D) cavity, steady cylindrical Couette flow, steady flow over a 2D circular cylinder, and steady and unsteady flows over a NACA0012 hydrofoil at different flow conditions. Results of the present solution are compared with the existing numerical and experimental results which show good agreement. To show the efficiency and accuracy of the solution methodology, the results are also compared with the developed second-order central-difference finite-volume lattice Boltzmann method and the compact finite-difference lattice Boltzmann method. It is shown that the present numerical scheme is robust, efficient, and accurate for solving steady and unsteady incompressible flows even at high Reynolds number flows.

Modified Non-linear Weights for Fifth-Order Weighted Essentially Non-oscillatory Schemes

This paper is concerned with fifth-order weighted essentially non-oscillatory (WENO) scheme with a new smoothness indicator. As the so-called WENO-JS scheme (Jiang and Shu in J Comput Phys 126:202---228, 1996) provides the third-order accuracy at critical points where the first and third order derivatives do not becomes zero simultaneously, several fifth-order WENO scheme have been developed through modifying the known smoothness indicators of WENO-JS. Recently a smoothness indicator based on $$L^1$$L1-norm has been proposed by Ha et al. (J Comput Phys 232:68---86, 2013) (denoted by WENO-NS). The aim of this paper is twofold. Firstly, we further improve the smoothness indicator of WENO-NS and secondly, using this measurement, we suggest new nonlinear weights by simplifying WENO-NS weights. The proposed WENO scheme provides the fifth-order accuracy, even at critical points. Some numerical experiments are provided to demonstrate that the present scheme performs better than other WENO schemes of the same order.

Delayed Detached Eddy Simulation of Airfoil Stall Flows Using High-Order Schemes

An advanced hybrid Reynolds-Averaged Navier–Stokes/large eddy simulation (RANS/LES) turbulence model delayed detached eddy simulation (DDES) is conducted in thispaper to investigate the dynamic stall flows over 3D NACA0012 airfoil at 17 deg, 26 deg, 45 deg, and 60 deg angle of attack (AOA). The spatially filtered unsteady 3D Navier–Stokes equations are solved using a fifth-order weighted essentially nonoscillatory (WENO) reconstruction with a low diffusion E-CUSP (LDE) scheme for the inviscid fluxes and a conservative fourth-order central differencing for the viscous terms. An implicit second-order time marching scheme with dual time stepping is employed to achieve high stability and convergency rate. A 3D flat plate is validated for the DDES model. For quantitative prediction of lift and drag of the stalled NACA0012 airfoil flows, the detached eddy simulation (DES) and DDES achieve much more accurate results than the Unsteady Reynolds-Averaged Navier–Stokes (URANS) simulation. In addition to the quantitative difference, the DES/DDES and URANS also obtain qualitatively very different unsteady stalled flows of NACA0012 airfoil with different vortical structures and frequencies. This may bring a significantly different prediction if those methods are used for fluid–structural interaction. For comparison purpose, a third-order WENO scheme with a second-order central differencing is also employed for the DDES stalled NACA0012 airfoil flows. Both the third- and fifth-order WENO schemes predict the stalled flow similarly for lift and drag at AOA less than 45 deg, while at AOA of 60 deg, the fifth-order WENO scheme shows better agreement with the experiment than the third-order WENO scheme. The high-order scheme of WENO 5 also resolves more small scales of flow structures than the second-order scheme. The prediction of the stalled airfoil flow using DDES with both the high-order scheme and second-order scheme is overall significantly more accurate than the URANS simulation.

WENO schemes on arbitrary unstructured meshes for laminar, transitional and turbulent flows

This paper presents the development and implementation of weighted-essentially-non-oscillatory (WENO) schemes for viscous flows on arbitrary unstructured grids. WENO schemes up to fifth-order accurate have been implemented in conjunction with hybrid and non-hybrid unstructured grids. The schemes are investigated with reference to numerical and experimental results for the Taylor–Green vortex, as well as for laminar and turbulent flows around a sphere, and the turbulent shock-wave boundary layer interaction flow problem. The results show that the accuracy of the schemes depends on the arbitrariness of shape and orientation of the unstructured mesh elements, as well as the compactness of directional stencils. The WENO schemes provide a more accurate numerical framework compared to second-order and third-order total variation diminishing (TVD) methods, however, the fifth-order version of the schemes is computationally too expensive to make the schemes practically usable. On the other hand, the third-order variant offers an excellent numerical framework in terms of accuracy and computational cost compared to the fifth-order WENO and second-order TVD schemes. Parallelisation of the CFD code (henceforth labelled as UCNS3D), where the schemes have been implemented, shows that the present methods offer very good scalable performance.