Non-oscillatory Interpolation Research Articles

Spherical Essentially Non-oscillatory (SENO) Interpolation

We develop two new ideas for interpolation on \(\mathbb {S}^2\). In this first part, we will introduce a simple interpolation method named Spherical Interpolation of orDER n (SIDER-n) that gives a \(C^{n}\) interpolant given \(n \ge 2\). The idea generalizes the construction of the Bézier curves developed for \(\mathbb {R}\). The second part incorporates the ENO philosophy and develops a new Spherical Essentially Non-Oscillatory (SENO) interpolation method. When the underlying curve on \(\mathbb {S}^2\) has kinks or sharp discontinuity in the higher derivatives, our proposed approach can reduce spurious oscillations in the high-order reconstruction. We will give multiple examples to demonstrate the accuracy and effectiveness of the proposed approaches.

ENO-based high-order data-bounded and constrained positivity-preserving interpolation

A number of key scientific computing applications that are based upon tensor-product grid constructions, such as numerical weather prediction (NWP) and combustion simulations, require property-preserving interpolation. Essentially non-oscillatory (ENO) interpolation is a classic example of such interpolation schemes. In the aforementioned application areas, property preservation often manifests itself as a requirement for either data boundedness or positivity preservation. For example, in NWP, one may have to interpolate between the grid on which the dynamics is calculated to a grid on which the physics is calculated (and back). Interpolating density or other key physical quantities without accounting for property preservation may lead to negative values that are nonphysical and result in inaccurate representations and/or interpretations of the physical data. Property-preserving interpolation is straightforward when used in the context of low-order numerical simulation methods. High-order property-preserving interpolation is, however, nontrivial, especially in the case where the interpolation points are not equispaced. In this paper, we demonstrate that it is possible to construct high-order interpolation methods that ensure either data boundedness or constrained positivity preservation. A novel feature of the algorithm is that the positivity-preserving interpolant is constrained; that is, the amount by which it exceeds the data values may be strictly controlled. The algorithm we have developed comes with theoretical estimates that provide sufficient conditions for data boundedness and constrained positivity preservation. We demonstrate the application of our algorithm on a collection of 1D and 2D numerical examples, and show that in all cases property preservation is respected.

A novel combination of mathematical modelling approaches for simulating nearly hypersonic, viscous, reacting magnetogasdynamic flows

This work focuses on the mathematical combination of magnetogasdynamics (MGD) and the Dunn and Kang chemical kinetic model with a multi-component Harten–Lax–van Leer Contact (HLLC) local Riemann solver using high-order interpolation schemes. Due to the nature of nearly hypersonic, unsteady, viscous, reacting MGD flows, the gas dissociates which is taken into account with 11 species and 26 chemical reactions along with the temperature dependent transport properties through the Wilke model. In the present work, the electromagnetic effects are considered by solving a Poisson equation simultaneously with the chemical reaction equations. The non-linear convective terms are treated numerically with third-, fifth- and seventh-order Weighted Essentially Non-Oscillatory (WENO) interpolation schemes along with second-, fourth- and sixth-order central finite difference schemes for the viscous terms. The combination of these numerical and physical models are analysed for stability and validated for benchmark problems such as (a) Hartmann flow as MGD validation test case and (b) the shock-tube problem for testing the chemical reaction model in which case the reaction rate controlling temperature suggested by Park has also been taken into account. Furthermore, a complex physical problem of an external aerodynamic flow over the cube for the Mach number varying from 1.25 to 3 have also been investigated, because a limited number of studies is available for understanding the flow behaviour of sharp cornered blunt bodies such as cubes. The knowledge of shock structures of these objects has become imperative to predict the atmospheric re-entry trajectories of de-orbiting CubeSats, space debris or fragments of launch vehicles. The main physical finding is that the shock structures are observed to be much more resolved compared to previous works, especially in the wake region. The knowledge gap is bridged by detailing the shock structures and showing the dependence of the shock standoff distance on total enthalpy of the flow.

A Semi-Lagrangian Computation of Front Speeds of G-equation in ABC and Kolmogorov Flows with Estimation via Ballistic Orbits

The Arnold-Beltrami-Childress (ABC) flow and the Kolmogorov flow are three dimensional periodic divergence free velocity fields that exhibit chaotic streamlines. We are interested in front speed enhancement in G-equation of turbulent combustion by large intensity ABC and Kolmogorov flows. We give a quantitative construction of the ballistic orbits of ABC and Kolmogorov flows, namely those with maximal large time asymptotic speeds in a coordinate direction. Thanks to the optimal control theory of G-equation (a convex but non-coercive Hamilton-Jacobi equation), the ballistic orbits serve as admissible trajectories for front speed estimates. To study the tightness of the estimates, we compute the front speeds of G-equation based on a semi-Lagrangian (SL) scheme with Strang splitting and weighted essentially non-oscillatory (WENO) interpolation. Time step size is chosen so that the Courant number grows sublinearly with the flow intensity. Numerical results show that the front speed growth rate in terms of the flow intensity may approach the analytical bounds from the ballistic orbits.

Nonlinear weighting process in ghost-cell immersed boundary methods for compressible flow

Computational challenges arise for the immersed boundary method (IBM) when dealing with compressible flow, where discontinuous and smoothly varying flow regions appear near the immersed boundary. The conventional ghost-cell IBM provides inaccurate results for smoothly varying regions when a low-order interpolation is used, or it suffers from a numerical instability for the discontinuous flow when a high-order interpolation is used. In this study, a new ghost-cell approach, nonlinear-weighted IBM (NWIBM), is developed to address the abovementioned issues. Inspired by a variety of weighted nonoscillatory interpolation methods, this work combines the high- and low-order polynomials during ghost-cell value estimation to enforce proper boundary conditions in the immersed boundary. A multidimensional smoothness indicator is designed to evaluate flow discontinuities. The nonlinear weighting obtained from the smoothness indicator makes the high-order polynomial dominant and the low-order polynomial negligible in the smoothly varying region, and vice versa in the discontinuous region. The enhanced performance and applicability of the proposed method were validated through various numerical tests in compressible flow. It was demonstrated that the NWIBM provides more stable and more accurate numerical solutions compared with conventional ghost-cell approaches.

A multi-resolution weighted compact nonlinear scheme for hyperbolic conservation laws

ABSTRACT A typical weighted compact nonlinear scheme (WCNS) uses a convex combination of several low-order polynomials approximated over selected candidate stencils of the same width, achieving non-oscillatory interpolation near discontinuities and high-order accuracy for smooth solutions. In this paper, we present a new multi-resolution fifth-order WCNS by making use of the information of polynomials on three nested central spatial sub-stencils having first-, third- and fifth-order accuracy, respectively. The new scheme is capable of obtaining high-order spatial interpolation in smooth regions, and it is characterised by the feature of gradually degrading from fifth-order down to first-order accuracy as large stencils deemed to be crossing strong discontinuities. The advantages of the present scheme include the superior resolution for high-wavenumber fluctuations and the flexibility of implementing different numerical flux functions.

An efficient high-order finite difference gas-kinetic scheme for the Euler and Navier–Stokes equations

Based on the temporal evolution of the Navier-Stokes gas distribution function and Weighted Essential Non-Oscillatory (WENO) interpolation, a high-order finite difference gas-kinetic scheme (FDGKS) is constructed. Different from the previous high-order finite volume gas-kinetic method [Li, Xu, and Fu, J. Comput. Phys. vol. 229, pp. 6715 (2010)], which uses a discontinuous initial reconstruction at the cell interface, the present scheme is a finite-difference one with a continuous flow distribution at the grid point.The time-accurate solution of the gas distribution function permits the FDGKS to be a one-step high-order scheme without multi-step Runge-Kutta temporal matching, which significantly reduces the computational time. Many numerical tests in solving one and two-dimensional Euler and Navier-Stokes equations demonstrate that FDGKS is a highly stable, accurate, and efficient scheme, which captures discontinuities without oscillations.

A parallel wavelet adaptive WENO scheme for 2D conservation laws

PurposeThe current work aims to present a parallel code using the open multi-processing (OpenMP) programming model for an adaptive multi-resolution high-order finite difference scheme for solving 2D conservation laws, comparing efficiencies obtained with a previous message passing interface formulation for the same serial scheme and considering the same type of 2D formulations laws.Design/methodology/approachThe serial version of the code is naturally suitable for parallelization because the spatial operator formulation is based on a splitting scheme per direction for which the flux components are numerically computed by a Lax–Friedrichs factorization independently for each row or column. High-order approximations for numerical fluxes are computed by the third-order essentially non-oscillatory (ENO) and fifth-order weighted essentially non-oscillatory (WENO) interpolation schemes, assuming sparse grids in each direction. The grid adaptivity is obtained by a cubic interpolating wavelet transform applied in each space dimension, associated to a threshold operator. Time is evolved by a third order TVD Runge–Kutta method.FindingsThe parallel formulation is implemented automatically at compiling time by the OpenMP library routines, being virtually transparent to the programmer. This over simplifies any concerns about managing and/or updating the adaptive grid when compared to what is necessary to be done when other parallel approaches are considered. Numerical simulations results and the large speedups obtained for the Euler equations in gas dynamics highlight the efficiency of the OpenMP approach.Research limitations/implicationsThe resulting speedups reflect the effectiveness of the OpenMP approach but are, to a large extension, limited by the hardware used (2 E5-2620 Intel Xeon processors, 6 cores, 2 threads/core, hyper-threading enabled). As the demand for OpenMP threads increases, the code starts to make explicit use of the second logical thread available in each E5-2620 processor core and efficiency drops. The speedup peak is reached near the possible maximum (24) at about 22, 23 threads. This peak reflects the hardware configuration and the true software limit should be located way beyond this value.Practical implicationsSo far no attempts have been made to parallelize other possible code segments (for instance, the ENO|-WENO-TVD code lines that process the different data components which could potentially push the speed up limit to higher values even further. The fact that the speedup peak is located close to the present hardware limit reflects the scalability properties of the OpenMP programming and of the splitting scheme as well. Consequently, it is likely that the speedup peak with the OpenMP approach for this kind of problem formulation will be close to the physical (and/or logical) limit of the hardware used.Social implicationsThis work is the result of a successful collaboration among researchers from two different institutions, one internationally well-known and with a long-term experience in applied mathematics for industrial applications and the other in a starting process of international academic insertion. In this way, this scientific partnership has the potential of promoting further knowledge exchange, involving students and other collaborators.Originality/valueThe proposed methodology (use of OpenMP programming model for the wavelet adaptive splitting scheme) is original and contributes to a very active research area in the past years, namely, adaptive methods for conservation laws and their parallel formulations, which is of great interest for the entire scientific community.

On a Power WENO Scheme with Improved Accuracy Near Discontinuities

This paper is devoted to the construction and analysis of a new prediction operator based on weighted essentially nonoscillatory (WENO) interpolation and capable of improving the accuracy near corner discontinuities in the point values and near jump discontinuities in the cell averages. The prediction operator presented is initially developed for the point values and then extended to the cell-average setting of Harten's multiresolution. In the cell averages we use the smoothness index proposed by Jiang and Shu [J. Comput. Phys., 126 (1996), pp. 202--228]. In the point values we use the one that was presented in [S. Amat and J. Ruiz, J. Sci. Comput., 71 (2017), pp. 1265--1302], where the authors show that the new smoothness index that they propose is better adapted for corner discontinuities. The inherent problem of WENO schemes and all the smoothness indexes available in the literature is that it is impossible to avoid the effect of the discontinuity when it is placed at the central interval of the stenci...

Conservative and Easily Implemented Finite Volume Semi-Lagrangian WENO Methods for 1D and 2D Hyperbolic Conservation Laws

The paper is devised to propose finite volume semi-Lagrange scheme for approximating linear and nonlinear hyperbolic conservation laws. Based on the idea of semi-Lagrangian scheme, we transform the integration of flux in time into the integration in space. Compared with the traditional semi-Lagrange scheme, the scheme devised here tries to directly evaluate the average fluxes along cell edges. It is this difference that makes the scheme in this paper simple to implement and easily extend to nonlinear cases. The procedure of evaluation of the average fluxes only depends on the high-order spatial interpolation. Hence the scheme can be implemented as long as the spatial interpolation is available, and no additional temporal discretization is needed. In this paper, the high-order spatial discretization is chosen to be the classical 5th-order weighted essentially non-oscillatory spatial interpolation. In the end, 1D and 2D numerical results show that this method is rather robust. In addition, to exhibit the numerical resolution and efficiency of the proposed scheme, the numerical solutions of the classical 5th-order WENO scheme combined with the 3rd-order Runge-Kutta temporal discretization (WENOJS) are chosen as the reference. We find that the scheme proposed in the paper generates comparable solutions with that of WENOJS, but with less CPU time.

A High Order Multi-Dimensional Characteristic Tracing Strategy for the Vlasov–Poisson System

In this paper, we consider a finite difference grid-based semi-Lagrangian approach for solving the Vlasov---Poisson (VP) system. Many of existing methods are based on dimensional splitting, which decouples the problem into solving linear advection problems, see Cheng and Knorr (J Comput Phys 22:330---351, 1976). However, such splitting is subject to the splitting error. If we consider multi-dimensional problems without splitting, difficulty arises in tracing characteristics with high order accuracy. Specifically, the evolution of characteristics is subject to the electric field which is determined globally from the distribution of particle density via Poisson's equation. In this paper, we propose a novel strategy of tracing characteristics high order in time via a two-stage multi-derivative prediction---correction approach and by using moment equations of the VP system. With the foot of characteristics being accurately located, we propose to use weighted essentially non-oscillatory interpolation to recover function values between grid points, therefore to update the solution at the next time level. The proposed method does not have time step restriction as the Eulerian approach and enjoys high order spatial and temporal accuracy. The performance of the proposed schemes are numerically demonstrated via classical test problems such as Landau damping and two stream instabilities.

A SEMI-LAGRANGIAN METHOD BASED ON WENO INTERPOLATION

Abstract. In this paper, a general Weighted Essentially Non-Oscillatory (WENO) interpolation is proposed and applied to asemi-Lagrangian method. The proposed method is based on theconservation law, and characteristic curves are used to completethe semi-Lagrangian method. Therefore, the proposed method sat-is es conservation of mass and is free of the CFL condition whichis a necessary condition for convergence.Using a several standardexamples, the proposed method is compared with the third orderStrong Stability Preserving (SSP) Runge-Kutta method to verifythe high-order accuracy. 1. IntroductionThe Weighted Essentially Non-Oscillatory (WENO) interpolations[8, 9, 10, 12, 13] are well known e ective high-order non-oscillatoryschemes based on Eulerian approach methods (grid based methods) asFinite Volume methods (FVM) or Finite Di erence methods (FDM).However, one of the drawbacks is that the methods are limited by theCourant-Friedrichs-Levy (CFL) condition which is a necessary conditionfor convergence.In order to resolve this challenging problem, a semi-Lagrangian methodconstructed by characteristic curves and grid points is introduced [1, 2, 3,4, 5, 14]. Our main idea is that time integration of a given conservationlaw (a partial di erential eqnarray to be solved) is changed into space in-tegration by using characteristic curves as the semi-Lagrangian method,and the Gaussian quadrature rule to obtain high-order accuracy. There-fore, it can not be guaranteed that the points of Gaussian quadraturerule computed by the characteristic curves are on regula grid points

Applying essentially non-oscillatory interpolation to controlled-source electromagnetic modelling*

Modelling and inversion of controlled-source electromagnetic (CSEM) fields requires accurate interpolation of modelled results near strong resistivity contrasts. There, simple linear interpolation may produce large errors, whereas higher-order interpolation may lead to oscillatory behaviour in the interpolated result. We propose to use the essentially non-oscillatory, piecewise polynomial interpolation scheme designed for piecewise smooth functions that contains discontinuities in the function itself or in its first or higher derivatives. The scheme uses a non-linear adaptive algorithm to select a set of interpolation points that represent the smoothest part of the function among the sets of neighbouring points. We present numerical examples to demonstrate the usefulness of the scheme. The first example shows that the essentially non-oscillatory interpolation (ENO) scheme better captures an isolated discontinuity. In the second example, we consider the case of sampling the electric field computed by a finite-volume CSEM code at a receiver location. In this example, the ENO interpolation performs quite well. However, the overall error is dominated by the discretization error. The other examples consider the comparison between sampling with essentially non-oscillatory interpolation and existing interpolation schemes. In these examples, essentially non-oscillatory interpolation provides more accurate results than standard interpolation, especially near discontinuities.

Numerical study of shock propagation and attenuation in narrow tubes including friction and heat losses

Numerical simulations of shock propagation and attenuation in narrow tubes are carried out using a one-dimensional approach. The discretization of the convective terms is based on the fifth-order weighted essentially non-oscillatory interpolation. The influence of the dissipative processes such as momentum and heat losses is investigated. Viscosity as well as heat losses are found to play a key role in the attenuation of the shock speed as well as the shock intensity in the long-time evolution, demonstrating the transition from a hyperbolic behavior towards a diffusive regime. Specifically, when only strong heat exchanges are considered, numerical tests, corroborated by a simple asymptotic analysis, showed a transition from a hyperbolic adiabatic regime to an isothermal regime. Furthermore, the influence of the scaling parameter ReD/4 L s , through the variation of the tube diameter, D, the viscous length scale, L s , and the Reynolds number on the shock propagation behavior is examined.

Implementation of a non-oscillatory and conservative scheme into magnetohydrodynamic equations

Abstract We present a magnetohydrodynamic (MHD) simulation technique with a new non-oscillatory and conservative interpolation scheme. Several high-resolution and stable numerical schemes have recently been proposed for solving the MHD equations. To apply the CIP scheme to the hydrodynamic equations, we need to add a certain diffusion term to suppress numerical oscillations at discontinuities. Although the TVD schemes can automatically avoid numerical oscillations, they are not appropriate for profiles with a local maximum or minimum, such as waves. To deal with the above problems, we implement a new non-oscillatory and conservative interpolation scheme in MHD simulations. Several numerical tests are carried out in order to compare our scheme with other recent high-resolution schemes. The numerical tests suggest that the present scheme can follow long-term evolution of both Alfvén waves and compressive shocks. The present scheme has been used for a numerical modeling of Alfvén waves in the solar wind, in which sinusoidal Alfvén waves decay into compressive sound waves that steepen into shocks.

Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics

The present paper introduces a class of finite volume schemes of increasing order of accuracy in space and time for hyperbolic systems that are in conservation form. The methods are specially suited for efficient implementation on structured meshes. The hyperbolic system is required to be non-stiff. This paper specifically focuses on Euler system that is used for modeling the flow of neutral fluids and the divergence-free, ideal magnetohydrodynamics (MHD) system that is used for large scale modeling of ionized plasmas. Efficient techniques for weighted essentially non-oscillatory (WENO) interpolation have been developed for finite volume reconstruction on structured meshes. We have shown that the most elegant and compact formulation of WENO reconstruction obtains when the interpolating functions are expressed in modal space. Explicit formulae have been provided for schemes having up to fourth order of spatial accuracy. Divergence-free evolution of magnetic fields requires the magnetic field components and their moments to be defined in the zone faces. We draw on a reconstruction strategy developed recently by the first author to show that a high order specification of the magnetic field components in zone-faces naturally furnishes an appropriately high order representation of the magnetic field within the zone. We also present a new formulation of the ADER (for Arbitrary Derivative Riemann Problem) schemes that relies on a local continuous space–time Galerkin formulation instead of the usual Cauchy–Kovalewski procedure. We call such schemes ADER-CG and show that a very elegant and compact formulation results when the scheme is formulated in modal space. Explicit formulae have been provided on structured meshes for ADER-CG schemes in three dimensions for all orders of accuracy that extend up to fourth order. Such ADER schemes have been used to temporally evolve the WENO-based spatial reconstruction. The resulting ADER-WENO schemes provide temporal accuracy that matches the spatial accuracy of the underlying WENO reconstruction. In this paper we have also provided a point-wise description of ADER-WENO schemes for divergence-free MHD in a fashion that facilitates computer implementation. The schemes reported here have all been implemented in the RIEMANN framework for computational astrophysics. All the methods presented have a one-step update, making them low-storage alternatives to the usual Runge–Kutta time-discretization. Their one-step update also makes them suitable building blocks for adaptive mesh refinement (AMR) calculations. We demonstrate that the ADER-WENO meet their design accuracies. Several stringent test problems of Euler flows and MHD flows are presented in one, two and three dimensions. Many of our test problems involve near infinite shocks in multiple dimensions and the higher order schemes are shown to perform very robustly and accurately under all conditions. It is shown that the increasing computational complexity with increasing order is handily offset by the increased accuracy of the scheme. The resulting ADER-WENO schemes are, therefore, very worthy alternatives to the standard second order schemes for compressible Euler and MHD flow.

A high-order WENO-Z finite difference based particle-source-in-cell method for computation of particle-laden flows with shocks

A high-order particle-source-in-cell (PSIC) algorithm is presented for the computation of the interaction between shocks, small scale structures, and liquid and/or solid particles in high-speed engineering applications. The improved high-order finite difference weighted essentially non-oscillatory (WENO-Z) method for solution of the hyperbolic conservation laws that govern the shocked carrier gas flow, lies at the heart of the algorithm. Finite sized particles are modeled as points and are traced in the Lagrangian frame. The physical coupling of particles in the Lagrangian frame and the gas in the Eulerian frame through momentum and energy exchange, is numerically treated through high-order interpolation and weighing. The centered high-order interpolation of the fluid properties to the particle location is shown to lead to numerical instability in shocked flow. An essentially non-oscillatory interpolation (ENO) scheme is devised for the coupling that improves stability. The ENO based algorithm is shown to be numerically stable and to accurately capture shocks, small flow features and particle dispersion. Both the carrier gas and the particles are updated in time without splitting with a third-order Runge–Kutta TVD method. One and two-dimensional computations of a shock moving into a particle cloud demonstrates the characteristics of the WENO-Z based PSIC method (PSIC/WENO-Z). The PSIC/WENO-Z computations are not only in excellent agreement with the numerical simulations with a third-order Rusanov based PSIC and physical experiments in [V. Boiko, V.P. Kiselev, S.P. Kiselev, A. Papyrin, S. Poplavsky, V. Fomin, Shock wave interaction with a cloud of particles, Shock Waves, 7 (1997) 275–285], but also show a significant improvement in the resolution of small scale structures. In two-dimensional simulations of the Mach 3 shock moving into forty thousand bronze particles arranged in the shape of a rectangle, the long time accuracy of the high-order method is demonstrated. The fifth-order PSIC/WENO-Z method with the fifth-order ENO interpolation scheme improves the small scale structure resolution over the third-order PSIC/WENO-Z method with a second-order central interpolation scheme. Preliminary analysis of the particle interaction with the flow structures shows that sharp particle material arms form on the side of the rectangular shape. The arms initially shield the particles from the accelerated flow behind the shock. A reflected compression wave, however, reshocks the particle arm from the shielded area and mixes the particles.

Visibility of point clouds and exploratory path planning in unkown environments

We present an algorithm for interpolating the visible portions of a point loud that are sampled from opaque objects in the environment. Our algorithm projects point clouds onto a sphere entered at the observing location and performs essentially non-oscillatory (ENO) interpolation of the projected data. Curvatures of the occluding objects can be approximated and used in many ways. We demonstrate how our visibility formulation can be incorporated into novel algorithms for mapping unknown environments with single or multiple observers, and target finding problems. A convergence proof is provided indicating suitability of our algorithm for some canonical types of environments. Various postprocessing optimization techniques are considered to obtain a more uniform exposure of the region along the path.

High-Order Accurate Numerical Method for Complex Flows

A numerical method employing high-order-accurate (higher than third) upwind discretizations for solving the compressible Navier-Stokes equations on structured grids is discussed. The inviscid fluxes are computed by a procedure based on a weighted essentially nonoscillatory interpolation of the characteristic variables and the Roe scheme. Application of the numerical method to a number of test cases of increasing complexity, that are prototypical for several of the key aspects of practical flows, demonstrates the accuracy and robustness of the method even when computing on distorted curvilinear grids. Significant reductions in computer time are possible when a second-order-accurate implicit Adams-Moulton scheme is employed for time integration. The combination of implicit time integration and high-order-accurate spatial discretization is shown to lead to significant savings in compute time as the grid resolution requirement is lowered and the time step can be increased.

Assessment of Very High Order of Accuracy in Implicit LES models

This paper looks at the use of high-resolution and very high-order methods for implicit large-eddy simulation (ILES), with the specific example of simulating the multicomponent two-dimensional single-mode Richtmyer–Meshkov instability for which experimental data is available. The two gases are air and SF6, making stringent demands on the models used in the code. The interface between the two gases is initialized with a simple sinusoidal perturbation over a wavelength of 59mm, and a shock of strength Mach 1.3 is passed through this interface. The main comparison is between the second-order monotone upwind-centered scheme for conservation law methods of van Leer (1979, “Towards the Ultimate Conservative Difference Scheme,” J. Comput. Phys. 32, pp. 101–136) and the current state-of-the-art weighted essentially nonoscillatory interpolation, which is presented to ninth order, concentrating on the effect on resolution of the instability on coarse grids. The higher-order methods as expected provide better resolved and more physical features than the second-order methods on the same grid resolution. While it is not possible to make a definitive statement, the simulations have indicated that the extra time required for the higher-order reconstruction is less than the time saved by being able to obtain the same or better accuracy at lower computational cost (fewer grid points). It should also be noted that all simulations give a good representation of the growth rate of the instability, comparing very favorably to the experimental results, and as such far better than the currently existing theoretical models. This serves to further indicate that the ILES approach is capable of providing accurately physical information despite the lack of any formal subgrid model.