Non-oscillatory Scheme Research Articles

On the learning of high order polynomial reconstructions for essentially non-oscillatory schemes

Approximation accuracy and convergence behavior are essential required properties for the computed numerical solution of differential equations. These requirements restrict the application of deep learning networks in the domain of scientific computing. Moreover, the recipe to create suitable synthetic data which can be used to train a good model is also not very clear. This study focuses on learning of third order essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstructions using classification neural networks with small data sets. In particular, this work (i) proposes a novel way to obtain a third order WENO reconstruction which can be posed as classification problem, (ii) gives simple and novel approach to sample data sets which are small but rich enough to inherit the latent feature of inter-spatial regularity information in the constructed data, (iii) it is established that sampling of train data sets impacts quantitatively as well as qualitatively the required accuracy and non-oscillatory properties of resulting ENO3 and WENO3 schemes, (iv) proposes to use a limiter based multi model to retain desired accuracy as well as non-oscillatory properties of the resulting numerical schemes. Computational results are given which established that learned networks perform well and retain the features of the reconstruction methods.

A new high-order RKDG method based on the TENO-THINC scheme for shock-capturing

In recent years, Runge-Kutta Discontinuous Galerkin (RKDG) methods have gained substantial attention in solving hyperbolic conservation laws, attributed to their high-order accuracy and adaptability to unstructured meshes. However, standard RKDG methods cannot capture discontinuities without oscillation unless they are supplemented with troubled cell indicators and limiters. Existing indicators, such as the total variation bounded (TVB) minmod indicator and the KXRCF indicator, typically depend on a critical parameter tied to the equation's solution, necessitating tuning for different cases. In terms of limiters, the popular limiters even fail to guarantee the high-order property in the smooth regions. The advent of Weighted Essentially Non-Oscillatory (WENO) schemes prompts the implementation of WENO limiters for RKDG methods, preserving high-order properties. However, WENO-family schemes exhibit significant numerical dissipation, potentially smearing small-scale flow structures. In this work, the Targeted Essentially Non-Oscillatory (TENO) indicator [1] is utilized, which leverages the nonlinear weighting strategy of the TENO scheme to separate high-wavenumber physical fluctuations and genuine discontinuities from smooth regions with a unified set of parameters. For troubled cells, a novel limiter is proposed for structured meshes, which combines a TENO scheme for resolving high-wavenumber physical fluctuations and a novel non-polynomial Tangent of Hyperbola for the INterface Capturing (THINC) scheme for resolving genuine discontinuities with extremely low numerical dissipation. Furthermore, the shifting between the TENO and THINC schemes is based on a new boundary variation diminishing (BVD) strategy, which only relies on compact neighborhoods and is significantly simpler than its predecessors. Meanwhile, a new strategy is proposed to ensure the consistency of the new limiter applied for 1D and 2D cases. A set of 1D and 2D benchmark cases including strong shockwaves and a broad range of flow length scales is simulated with uniform meshes for 1D cases and structured quadrilateral meshes for 2D cases to demonstrate the performance of the new numerical scheme. The indicator does not activate any limiters in the accuracy test cases to ensure the high-order property of the whole numerical scheme. Other cases with discontinuities demonstrate the low-dissipation properties of the new limiter in comparison to the simple WENO limiter.

Numerical investigation on the blade vortex interaction noise reduction using higher harmonic control

Accurately predicting the formation, evolution, and breakdown of helicopter blade tip vortex is crucial for simulating the Blade-Vortex Interaction (BVI) phenomenon. Firstly, the high-order Perturbed polynomial reconstructed Targeted Essentially Non-Oscillatory (TENO-P) scheme proposed by our research group is employed to improve the resolution of the helicopter rotor flowfield solver. The TENO-P scheme, building on the fifth-order TENO5 scheme, achieves one-order of accuracy improvement by adaptively adjusting the values of the free-parameter introduced by perturbed polynomial reconstruction. Subsequently, the AH-1 helicopter model rotor undergoing blade-vortex interaction is analyzed using the improved rotor flowfield solver and the Farassat-1A formula. The implementation of the TENO-P scheme notably enhances the resolution of the rotor flowfield solver in resolving the blade tip vortex structures, and the predicted noise results are in good agreement with the experimental data. Finally, the alteration of the BVI noise and its reduction mechanism of the AH-1 model rotor under different Higher Harmonic Controls (HHC) are analyzed. The findings show that the phase modulation margin increases with a decrease in harmonic order, and the noise reduction effect significantly improves as well. The negative component of the higher harmonic control is beneficial for reducing BVI noise while the positive component of the higher harmonic control exacerbates the BVI noise. The HHC reduces the collective pitch angle in the region where the tip vortex is about to be disturbed, decreasing the strength of the blade tip vortex in this region. This reduction weakens the strength of the parallel and near-parallel interaction on the advancing side, leads to the reduction of the positive peak impulsive of the BVI noise, and consequently lowers the intensity of the BVI noise.

High-order oscillation-eliminating Hermite WENO method for hyperbolic conservation laws

This paper proposes high-order accurate, oscillation-eliminating, Hermite weighted essentially non-oscillatory (OE-HWENO) finite volume schemes for hyperbolic conservation laws, motivated by the oscillation-eliminating (OE) discontinuous Galerkin schemes recently proposed in [M. Peng, Z. Sun, and K. Wu, 2024 [30]]. The OE-HWENO schemes incorporate an OE procedure after each Runge–Kutta stage, by dampening the first-order moments of the HWENO solution to suppress spurious oscillations without any problem-dependent parameter. The OE procedure acts as a moment filter and is derived from the solution operator of a novel damping equation, which is exactly solved without any discretization. Thanks to this distinctive feature, the OE-HWENO method remains stable with a normal CFL number, even for strong shocks resulting in highly stiff damping terms. To ensure the essentially non-oscillatory property of the OE-HWENO method across problems with varying scales and wave speeds, we design a scale-invariant and evolution-invariant damping equation and propose a generic dimensionless transformation for HWENO reconstruction. The OE-HWENO method offers several notable advantages over existing HWENO methods. First, the OE procedure is highly efficient and straightforward to implement, requiring only simple multiplication of first-order moments by a damping factor. Furthermore, we rigorously prove that the OE procedure maintains the high-order accuracy and local compactness of the original HWENO schemes and demonstrate that it does not compromise the spectral properties via the approximate dispersion relation for smooth solutions. Notably, the proposed OE procedure is non-intrusive, enabling seamless integration as an independent module into existing HWENO codes. Finally, we rigorously analyze the bound-preserving (BP) property of the OE-HWENO method using the optimal cell average decomposition approach [S. Cui, S. Ding, and K. Wu, 2024 [8]], which relaxes the theoretical BP constraint for time step-size and reduces the number of decomposition points, thereby further enhancing efficiency. Extensive benchmarks validate the accuracy, efficiency, high resolution, and robustness of the OE-HWENO method.

A family of TENOA-THINC-MOOD schemes based on diffuse-interface method for compressible multiphase flows

In this work, we develop a family of hybrid algorithms to simulate compressible multi-phase flows by solving the quasi-conservative five-equation model. The general numerical framework utilizes targeted essentially non-oscillatory schemes with adaptive dissipation (TENOA) to boost the resolution of high-frequency waves, whereas the Tangent of Hyperbola for INterface Capturing (THINC) scheme is activated near the shock and contact discontinuities as well as material interfaces to maintain their sharpness, based on a novel indicator. Also, the newly derived THINC reconstruction scheme is both simpler and better at preserving the flow symmetry. The robustness of our numerical approach is further fortified through the adaptation of a modified a posteriori multidimensional optimal order detection (MOOD) technique. Moreover, to mitigate carbuncle phenomenon encountered in two-dimensional simulations, a hybrid Riemann solver is also proposed based on the low Mach modification around shock for the Harten-Lax-van Leer contact (HLLC) approximate Riemann solver. Numerical results of the challenging one- and two-dimensional benchmark cases demonstrate the superiority of the proposed methods regarding oscillation-free, interface-sharpening, low-dissipation and robustness properties.

High order conservative finite difference WENO scheme for three-temperature radiation hydrodynamics

The three-temperature (3-T) radiation hydrodynamics (RH) equations play an important role in the high-energy-density-physics fields, such as astrophysics and inertial confinement fusion (ICF). It describes the interaction between radiation and high-energy-density plasmas including electron and ion in the assumption that radiation, electron and ion are in their own equilibrium state, which means they can be characterized by their own temperatures. The 3-T RH system consists of the density, momentum and three internal energy (electron, ion and radiation) equations. In this paper, we propose a high order conservative finite difference weighted essentially non-oscillatory (WENO) scheme solving one-dimensional (1D) and two-dimensional (2D) 3-T RH equations respectively. Following our previous paper [7], we introduce the three new energy variables, and then design a finite difference scheme with both the conservative property and arbitrary high order accuracy. Based on the WENO interpolation and the strong stability preserving (SSP) high order time discretizations, taken as an example, we design a class of fifth order conservative finite difference schemes in space and third order in time. Compared with the Lagrangian method we proposed in [7], which can only reach second order accuracy for 2D 3-T RH equations if straight-line edged meshes are used, the finite difference scheme can be easily designed to arbitrary high order accuracy for multi-dimensional 3-T RH equations. The finite difference formulation is also much less expensive in multi-dimensions than finite volume schemes used in [7]. Furthermore, our method can handle fluids with large deformation easily. Numerous 1D and 2D numerical examples are presented to verify the desired properties of the high order finite difference WENO schemes such as high order accuracy, non-oscillation, conservation and adaptation to severely distorted single-material radiation hydrodynamics problems.

Sixth-order perturbed WENO interpolation-based AWENO and WCNS-E schemes for hyperbolic conservation laws

The weighted essentially non-oscillatory (WENO) interpolation-based schemes (the alternative WENO (AWENO) scheme and the explicit weighted compact nonlinear (WCNS-E) scheme) are limited to the fifth-order despite the sixth-order Taylor expansion of the numerical flux used. This order reduction is due to the fifth-order accuracy inherent in the WENO interpolation. We investigate the perturbed WENO interpolation with a free parameter and affine-invariant WENO weights to recover sixth-order accuracy in smooth regions. A cutoff function of the free parameter with a threshold, determined by approximate dispersion relation analysis, is applied to enhance the ENO property. The proposed schemes perform better in accuracy, dissipation, resolution, shock-capturing, and efficiency in 1D and 2D benchmark problems.

Increasingly high-order hybrid multi-resolution WENO schemes in multi-dimensions

In this paper, a new type of increasingly high-order hybrid multi-resolution weighted essentially non-oscillatory (HMR-WENO) schemes is presented in the finite difference framework for solving hyperbolic conservation laws in one, two, and three dimensions. Based on the reconstruction polynomials defined on the one-point, three-point, five-point, seven-point, and nine-point spatial stencils, we reconstruct one zeroth degree reconstruction polynomial, one quadratic reconstruction polynomial, one quartic reconstruction polynomial, one sextic reconstruction polynomial, one octave reconstruction polynomial, and their derivative polynomials together with a new hierarchical bisection method to design a series of new troubled cell indicators which can precisely find all extreme points of associated unequal degree reconstruction polynomials located inside the smallest interval in one dimension. The new troubled cell indicators do not introduce any manual parameters related to different problems. The new hybrid methodology is divided into two parts: if all extreme points of the reconstruction polynomials are nonexistent or outside the smallest interval, the target cell is not a troubled cell and the simple linear upwind schemes are utilized to obtain high-order approximations. Otherwise, the target cell is a troubled cell and the MR-WENO spatial reconstruction procedures with excellent shock-capture ability are adopted. Then a series of HMR-WENO schemes are proposed by using these new troubled cell indicators, which can be easily expanded to arbitrarily high-order accuracies in multi-dimensions. The main benefits of these HMR-WENO schemes are their efficiency, since they could save about 22%-75% CPU time than that of the same order MR-WENO schemes when simulating some benchmark examples in multi-dimensions.

Weighted compact nonlinear scheme based on smooth scale separation for self-adaptive turbulence eddy simulations

Multiscale compressible turbulent flow is widely present in the aerospace field, and the co-existence of shock wave discontinuities and multi-scale turbulence in the flow field poses significant challenges for high-fidelity numerical simulation. To design high-order nonlinear scheme to improve the accuracy of complex turbulence simulation with stable shock wave capture, a weighted compact nonlinear scheme – targeted essentially non-oscillatory (WCNS-T) scheme is developed based on targeted essentially non-oscillatory scheme within the WCNS framework. Also, an improved variant of WCNS-T with smooth scale separation (denoted as WCNS-ST) is proposed from the perspective of both numerical convergence and accuracy. The numerical properties of different nonlinear weights are discussed in classical numerical problems, including spectral properties analysis, and two-dimensional Riemann problem. Then, the newly developed WCNS-ST scheme based on smooth scale separation was used to combine a unified turbulence model called Self-Adaptive Turbulence Eddy Simulation (SATES), which is applied to simulate subsonic flow past a 30P30N airfoil and supersonic base flow. The good convergence and numerical results are obtained using the newly developed WCNS-ST scheme, whereas WCNS-T shows worse convergence. In addition, adaptive dissipation control is further introduced for the WCNS-ST scheme, and the new variant (denoted as WCNS-STA) still shows good residual convergence and predicts more accurate results due to low numerical dissipation. The above results indicate that the proposed smooth scale separation and its WCNS-STA scheme have a good performance in SATES simulations of multiscale flow problems, making it more potential for complex engineering applications.

Line-Based High-Order Hybrid Scheme on Unstructured Grids for Compressible Flows

This study explores the use of line-based finite difference techniques on unstructured grids employing high-order stencils for practical aerodynamic applications. The primary focus is on finite-difference explicit and compact spatial discretization up to sixth order. Problems studied include compressible flows containing discontinuities, which necessitated the use of a hybrid finite difference–finite volume approach, where the former is shown to be effective and accurate for smooth regions, while the latter is robust in capturing discontinuities and shocks. To differentiate between discontinuous and smooth zones, two distinct shock indicators are used, one based on pressure gradients and the other on the divergence of velocity. To achieve faster convergence, spatial discretization is applied with line-based alternating direction implicit (ADI) time integration. A key focus of this work is to demonstrate the proposed methodology in flow scenarios relevant to aircraft applications that also test various aspects of the methodology. Consequently, the cases studied are isentropic vortex convection, standing normal shock, transonic NACA0012 airfoil, supersonic cylinder, unsteady flow past tandem circular cylinders, and supersonic flow past a forward-facing step. Comparisons were also done against an existing line-based finite-volume technique for unstructured grids that employed Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and fifth-order Weighted Essentially NonOscillatory (WENO5) spatial reconstruction schemes. It is observed that the hybrid finite difference–finite volume approach performed well with excellent agreement against available results, sometimes with far fewer degrees of freedom. Furthermore, the implicit ADI scheme is effective over unstructured grids and provides a three-time speed-up over the traditional Runge–Kutta fourth-order scheme.

A resolution-enhanced seventh-order weighted essentially non-oscillatory scheme based on non-polynomial reconstructions for solving hyperbolic conservation laws

In high-resolution numerical simulations of flows characterized by both multiscale turbulence and discontinuities, the conflict between spectral characteristics and stability becomes increasingly pronounced as the order of accuracy improves. To address this challenge, we proposed a novel seventh-order weighted essentially non-oscillatory scheme (WENO-K7). This scheme utilizes non-polynomial reconstructions by incorporating kriging interpolation and Gaussian exponential function. Then, a hyper-parameter associated with the Gaussian function is adaptively optimized to achieve higher convergence orders on sub-stencils, reducing numerical errors on global stencils. Additionally, a criterion based on monotone interpolations is devised to automatically identify problematic hyper-parameters, facilitating the transition from non-polynomial to polynomial reconstructions near discontinuities and preserving the essentially non-oscillatory property. Compared to the conventional seventh-order WENO-Z7 scheme, WENO-K7 scheme exhibits smaller computational error and reduced numerical dissipation in smooth regions while maintaining non-oscillatory and high-resolution capabilities around discontinuities. Results from various one- and two-dimensional benchmark cases demonstrate that the proposed WENO-K7 scheme outperforms the widely used WENO-Z7 scheme with only a 12% increase in computational cost. Moreover, the WENO-K7 scheme shares the same sub-stencils as the WENO-Z7 scheme, making it easily applicable to other variants of seventh-order WENO schemes and enhancing their spectral characteristics.

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.

A high-resolution numerical investigation of unsteady wake vortices for coaxial rotors in hover

High-resolution numerical simulations for wake vortical flows have long been a challenge in rotor aerodynamics. A novel spectrum-optimized sixth-order Weighted Essentially Non-Oscillatory (WENO) scheme is proposed to discretize inviscid fluxes on moving overset grids, and the Improved Delayed Detached Eddy Simulation (IDDES) is employed to resolve turbulent vortices. The integration of these methods facilitates a comprehensive numerical investigation into the unsteady vortical flows over coaxial rotors in hover. The results highlight the substantial improvement in numerical resolution, in terms of both spatial structure and temporal evolution of unsteady multiscale wake vortices. Coaxial rotors in hover manifest three primary scales of wake vortex structures: (A) the helical evolution of primary blade tip vortices and the periodic occurrence of strong Blade-Vortex Interactions (BVI); (B) the continuous shedding of small-scale horseshoe-shaped vortices from the trailing edges of rotor blades, forming the vortex sheets; (C) the emergence of small-scale secondary vortex braids induced by interactions between rotor tip vortices and the vortex sheets. These vortex structures and their interactions cause high-frequency oscillations in rotor disk loads and induce unsteady perturbations in the local flow field. Interactions among these primary vortices, coupled with the generation of secondary vortices, result in the dissipation, distortion, and breakup of the rotor tip vortices, ultimately forming a vortex soup. Notably, a substantial quantity of seemingly weak small-scale secondary vortex braids significantly contribute to energy dissipation during the evolution of wake vortices for coaxial rotors in hover.

Convergence towards High-Speed Steady States Using High-Order Accurate Shock-Capturing Schemes

Creating time-marching unsteady governing equations for a steady state in high-speed flows is not a trivial task. Residue convergence in time cannot be achieved when using most low- and high-order spatial discretization schemes. Recently, high-order, weighted, essentially non-oscillatory schemes have been specially designed for steady-state simulations. They have been shown to be capable of achieving machine precision residues when simulating the Euler equations under canonical coordinates. In the present work, we review these schemes and show that they can also achieve machine residues when simulating the Navier–Stokes equations under generalized coordinates. This is carried out by considering three supersonic flows of perfect fluids, namely the flow upstream a cylinder, the flow over a blunt wedge, and the flow over a compression ramp.

Numerical simulation of 3-D seismic wave based on alternative flux finite-difference WENO scheme

SUMMARY High-frequency non-physical oscillations may occur due to shock waves in seismic wavefield and dynamic rupture simulation. In this study, we introduced the alternative flux finite-difference weighted essentially non-oscillatory scheme to address potential shock wave issues in computational seismology effectively. The wavefield of the body-fitted curvilinear domain was accurately computed through conservative grid mapping, ensuring accurate implementation of free surface boundary conditions on irregular surfaces using characteristic boundary conditions and minimizing artificial boundary reflections with exponential decay absorbing layers. Finally, we compared our scheme with the GRTM for flat surfaces and the CGFDM3D-EQR for irregular surfaces to demonstrate its correctness and accuracy, and validated its non-oscillatory characteristics. The aforementioned scheme is anticipated to assume a significant function in simulating more intricate seismic wavefields or dynamic ruptures.

Comparison of high-order numerical methodologies for the simulation of the supersonic Taylor–Green vortex flow

This work presents a comparison of several high-order numerical methodologies for simulating shock/turbulence interactions based on the supersonic Taylor–Green vortex flow, considering a Reynolds number of 1600 and a Mach number of 1.25. The numerical schemes considered include high-order Finite Difference, Targeted Essentially Non-Oscillatory, Discontinuous Galerkin, and Spectral Difference schemes. The shock capturing methods include high-order filtering, localized artificial diffusivity, non-oscillatory numerical fluxes, and local low-order switching. The ability of the various high-order numerical methodologies to both capture shocks and represent accurately the development of turbulent vortices is assessed.

Fourth-Order Conservative Non-splitting Semi-Lagrangian Hermite WENO Schemes for Kinetic and Fluid Simulations

AbstractWe present fourth-order conservative non-splitting semi-Lagrangian (SL) Hermite essentially non-oscillatory (HWENO) schemes for linear transport equations with applications for nonlinear problems including the Vlasov–Poisson system, the guiding center Vlasov model, and the incompressible Euler equations in the vorticity-stream function formulation. The proposed SL HWENO schemes combine a weak formulation of the characteristic Galerkin method with two newly constructed HWENO reconstruction methods. The new HWENO reconstructions are meticulously designed to strike a delicate balance between curbing numerical oscillation and introducing excessive dissipation. Mass conservation naturally holds due to the weak formulation of the semi-Lagrangian discontinuous Galerkin method and the design of the HWENO reconstructions. We apply a positivity-preserving limiter to maintain the positivity of numerical solutions when needed. Abundant benchmark tests are performed to verify the effectiveness of the proposed SL HWENO schemes.

WENO schemes using optimized third order fuzzy weight limiter functions

Abstract This work presents an improved version of non-linear weight limiters to obtain third order non-oscillatory WENO scheme. The construction of this modified weight limiter is based on fuzzy inference system, which is a knowledge based rule system. The linear combination of overlapped basis functions is used to achieve the optimized weight limiters by exploring the linguistics hedges operator on the basis functions. The WENO scheme using optimized weight limiter achieves third order of accuracy and gives higher resolution to discontinuities compared to other established third order WENO schemes.

A new type of modified MR-WENO schemes with new troubled cell indicators for solving hyperbolic conservation laws in multi-dimensions

In this paper, a new type of increasingly high-order modified multi-resolution weighted essentially non-oscillatory (MMR-WENO) schemes with new troubled cell indicators is designed in the finite difference framework for solving hyperbolic conservation laws in one, two, and three dimensions. It is a first time to design new troubled cell indicators that based on two high-degree reconstruction polynomials, which are defined on the three-point, five-point, seven-point, and nine-point spatial stencils, respectively. Such new troubled cell indicators can automatically identify the discontinuous solutions without manually adjusting the parameters related to the problems. Subsequently, a series of MMR-WENO schemes are designed by using these new troubled cell indicators, which use the simple linear upwind schemes in smooth areas and the sophisticated MR-WENO schemes in discontinuous areas, thus achieving the goal of inheriting the excellent characteristics of original MR-WENO schemes while reducing computational costs. The new modified methodology is divided into two parts: if all extreme points of two reconstruction polynomials defined on the big spatial stencil are outward the smallest interval [xi−1/2,xi+1/2], the numerical flux is straightforwardly approximated by a high-degree reconstruction polynomial and the approximation has a high-order accuracy. Otherwise, the high-order MR-WENO spatial reconstruction procedures are adopted. The main benefits of new MMR-WENO schemes are their robustness and efficiency in comparison to original MR-WENO schemes, since the MMR-WENO schemes could save at most 63%-79% CPU time than the same-order MR-WENO schemes do for some numerical examples.

On the accuracy of shock-capturing schemes when calculating Cauchy problems with periodic discontinuous initial data

Abstract We study the accuracy of shock-capturing schemes for the shallow water Cauchy problems with piecewise smooth discontinuous initial data. We consider the second order balance-characteristic (CABARETM) scheme, the third order finite-difference Rusanov–Burstein–Mirin (RBM) scheme and the fifth order in space, the third order in time weighted essentially non-oscillatory (WENO5) scheme. We have shown that the maximum loss of accuracy occurs in the centered rarefaction waves of the exact solutions, where all these schemes have the first order of convergence and fairly close values of the numerical disbalances (errors), regardless of their formal approximation order on the smooth solutions. In the same time, inside the shock influence areas the considered schemes can have different convergence orders and, as a result, significantly different accuracy. In particular, when solving the Cauchy problem with periodic initial data, when the exact solution has no centered rarefaction waves, the RBM scheme has a significantly higher accuracy inside the shock influence areas, compared to the CABARETM and WENO5 schemes. It means that the combined scheme, in which the RBM scheme is a basic scheme and the CABARETM scheme is an internal one, can be effectively used to compute weak solutions of such type Cauchy problems.