Order Spatial Accuracy Research Articles

WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions

The paper extends weighted essentially non-oscillatory (WENO) methods to three dimensional mixed-element unstructured meshes, comprising tetrahedral, hexahedral, prismatic and pyramidal elements. Numerical results illustrate the convergence rates and non-oscillatory properties of the schemes for various smooth and discontinuous solutions test cases and the compressible Euler equations on various types of grids. Schemes of up to fifth order of spatial accuracy are considered.

Modelling of impulse loading in high‐temperature superconductors

PurposeThe aim of this paper is to access performance of existing computational techniques to model strongly non‐linear field diffusion problems.Design/methodology/approachMultidimensional application of a finite volume front‐fixing method to various front‐type problems with moving boundaries and non‐linear material properties is discussed. Advantages and implementation problems of the technique are highlighted by comparing the front‐fixing method with computations using fixed grids. Particular attention is focused on conservation properties of the algorithm and accurate solutions close to the moving boundaries. The algorithm is tested using analytical solutions of diffusion problems with cylindrical symmetry with both spatial and temporal accuracy analysed.FindingsSeveral advantages are identified in using a front‐fixing method for modelling of impulse phenomena in high‐temperature superconductors (HTS), namely high accuracy can be obtained with a small number of grid points, and standard numerical methods for convection problems with diffusion can be utilised. Approximately, first order of spatial accuracy is found for all methods (stationary or mobile grids) for 2D problems with impulse events. Nevertheless, errors resulting from a front‐fixing technique are much smaller in comparison with fixed grids. Fractional steps method is proved to be an effective algorithm for solving the equations obtained. A symmetrisation procedure has to be introduced to eliminate a directional bias for a standard asymmetric split in diffusion processes.Originality/valueThis paper for the first time compares in detail advantages and implementation complications of a front‐fixing method when applied to the front‐type field diffusion problems common to HTS. Particular attention is paid to accurate solutions in the region close to the moving front where rapid changes in material properties are responsible for large computational errors.

Numerical Schemes with High Spatial Accuracy for a Variable-Order Anomalous Subdiffusion Equation

In this paper, we consider a variable-order anomalous subdiffusion equation. A numerical scheme with first order temporal accuracy and fourth order spatial accuracy for the equation is proposed. The convergence, stability, and solvability of the numerical scheme are discussed via the technique of Fourier analysis. Another improved numerical scheme with second order temporal accuracy and fourth order spatial accuracy is also proposed. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

Finite difference modelling of rupture propagation with strong velocity-weakening friction

SUMMARY We incorporate rate- and state-dependent friction in explicit finite difference (FD) simulations of mode II dynamic ruptures in elastic media, using the Mimetic Operators Split-Node (MOSN) method, with adjustable order of spatial accuracy (second-, fourth- or mixed-order accurate), including an option that is fourth-order accurate at the fault discontinuity as well as in the elastic volume. At fault points, the rate and state equations combined with the spatially discretized momentum conservation equations form a coupled system of ordinary differential equations (ODEs) for slip velocity and state variable. As a consequence of the rapid damping of velocity perturbations due to the direct effect, this system exhibits numerical stiffness that is inversely proportional to velocity squared. Approximate solutions to this velocity-state system are achieved by two different implicit schemes: (i) a fourth-order Rosenbrock integration of the full system using multiple substeps and (ii) low order integrations (backward Euler and trapezoidal) of the velocity equation, time-staggered with analytic integration of the state equation under the approximation of constant slip velocity over the time step. In assessing the numerical schemes, we use three test problems: ruptures with frictional resistance controlled by (i) a slip evolution law with strong velocity-weakening behaviour at high slip rates, representing thermal weakening due to flash heating of microscopic asperity contacts, (ii) the classic (low-velocity) slip evolution law and (iii) the classic aging evolution law. A convergence analysis is carried out using reference solutions from a spectral boundary integral equation method (BIEM) (a method restricted to homogeneous media, with nominal spectral accuracy in space and second-order accuracy in time for smooth solutions). Errors are measured by root-mean-square differences of fault-plane time histories (slip, slip rate, traction and state). MOSN shows essentially the same convergence rates as BIEM: second-order convergence for slip and state-variable misfits, with slower (but at least first-order) convergence for slip rates and tractions. For a given grid spacing, fourth-order MOSN is as accurate as BIEM for all variables except slip-rate. MOSN-Rosenbrock nominally has fourth-order temporal accuracy for the fault-plane velocity-state ODE integration (compared to lower-order accuracy for the other two MOSN schemes) and therefore provides an important theoretical benchmark. However, it is sensitive to details of the elastic calculation scheme and occasionally its adaptive substepping performs poorly, leading to large excursions from the reference solution. In contrast, MOSN-trapezoidal is robust and reliable, much easier to implement than MOSN-Rosenbrock, and in all cases achieves precision as good as the latter without recourse to substepping. MOSN-Euler has the same advantages as MOSN-trapezoidal, except that its nominal first-order temporal accuracy ultimately leads to larger errors in slip and state variable compared with the higher-order MOSN schemes at sufficiently small grid spacings and time steps.

Performance of very‐high‐order upwind schemes for DNS of compressible wall‐turbulence

AbstractThe purpose of the present paper is to evaluate very‐high‐order upwind schemes for the direct numerical simulation (DNS) of compressible wall‐turbulence. We study upwind‐biased (UW) and weighted essentially nonoscillatory (WENO) schemes of increasingly higher order‐of‐accuracy (J. Comp. Phys. 2000; 160:405–452), extended up to WENO17 (AIAA Paper 2009‐1612, 2009). Analysis of the advection–diffusion equation, both as Δx→0 (consistency), and for fixed finite cell‐Reynolds‐number ReΔx (grid‐resolution), indicates that the very‐high‐order upwind schemes have satisfactory resolution in terms of points‐per‐wavelength (PPW). Computational results for compressible channel flow (Re∈[180, 230]; M̄CL∈[0.35, 1.5]) are examined to assess the influence of the spatial order of accuracy and the computational grid‐resolution on predicted turbulence statistics, by comparison with existing compressible and incompressible DNS databases. Despite the use of baseline O(Δt2) time‐integration and O(Δx2) discretization of the viscous terms, comparative studies of various orders‐of‐accuracy for the convective terms demonstrate that very‐high‐order upwind schemes can reproduce all the DNS details obtained by pseudospectral schemes, on computational grids of only slightly higher density. Copyright © 2009 John Wiley & Sons, Ltd.

A finite volume method parallelization for the simulation of free surface shallow water flows

We construct a parallel algorithm, suitable for distributed memory architectures, of an explicit shock-capturing finite volume method for solving the two-dimensional shallow water equations. The finite volume method is based on the very popular approximate Riemann solver of Roe and is extended to second order spatial accuracy by an appropriate TVD technique. The parallel code is applied to distributed memory architectures using domain decomposition techniques and we investigate its performance on a grid computer and on a Distributed Shared Memory supercomputer. The effectiveness of the parallel algorithm is considered for specific benchmark test cases. The performance of the realization measured in terms of execution time and speedup factors reveals the efficiency of the implementation.

Time-Derivative Preconditioning Methods for Multicomponent Flows—Part II: Two-Dimensional Applications

A time-derivative preconditioned system of equations suitable for the numerical simulation of multicomponent/multiphase inviscid flows at all speeds was described in Part I of this paper. The system was shown to be hyperbolic in time and remain well conditioned in the incompressible limit, allowing time marching numerical methods to remain an efficient solution strategy. Application of conservative numerical methods to multicomponent flows containing sharp fluid interfaces was shown to generate nonphysical pressure and velocity oscillations across the contact surface, which separates the fluid components. It was demonstrated using the one-dimensional Riemann problem that these oscillations may lead to stability problems when the interface separates fluids with large density ratios, such as water and air. The effect of which leads to the requirement of small physical time steps and slow subiteration convergence for the implicit time marching numerical method. Alternatively, the nonconservative and hybrid formulations developed by the present authors were shown to eliminate this nonphysical behavior. While the nonconservative method did not converge to the correct weak solution for flow containing shocks, the hybrid method was able to capture the physically correct entropy solution and converge to the exact solution of the Riemann problem as the grid is refined. In Part II of this paper, the conservative, nonconservative, and hybrid formulations described in Part I are implemented within a two-dimensional structured body-fitted overset grid solver, and a study of two unsteady flow applications is reported. In the first application, a multiphase cavitating flow around a NACA0015 hydrofoil contained in a channel is solved, and sensitivity to the cavitation number and the spatial order of accuracy of the discretization are discussed. Next, the interaction of a shock moving in air with a cylindrical bubble of another fluid is analyzed. In the first case, the cylindrical bubble is filled with helium gas, and both the conservative and hybrid approaches perform similarly. In the second case, the bubble is filled with water and the conservative method fails to maintain numerical stability. The performance of the hybrid method is shown to be unchanged when the gas is replaced with a liquid, demonstrating the robustness and accuracy of the hybrid approach.

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.

High order conservative finite difference scheme for variable density low Mach number turbulent flows

The high order conservative finite difference scheme of Morinishi et al. [Y. Morinishi, O.V. Vasilyev, T. Ogi, Fully conservative finite difference scheme in cylindrical coordinates for incompressible flow simulations, J. Comput. Phys. 197 (2004) 686] is extended to simulate variable density flows in complex geometries with cylindrical or cartesian non-uniform meshes. The formulation discretely conserves mass, momentum, and kinetic energy in a periodic domain. In the presence of walls, boundary conditions that ensure primary conservation have been derived, while secondary conservation is shown to remain satisfactory. In the case of cylindrical coordinates, it is desirable to increase the order of accuracy of the convective term in the radial direction, where most gradients are often found. A straightforward centerline treatment is employed, leading to good accuracy as well as satisfactory robustness. A similar strategy is introduced to increase the order of accuracy of the viscous terms. The overall numerical scheme obtained is highly suitable for the simulation of reactive turbulent flows in realistic geometries, for it combines arbitrarily high order of accuracy, discrete conservation of mass, momentum, and energy with consistent boundary conditions. This numerical methodology is used to simulate a series of canonical turbulent flows ranging from isotropic turbulence to a variable density round jet. Both direct numerical simulation (DNS) and large eddy simulation (LES) results are presented. It is observed that higher order spatial accuracy can improve significantly the quality of the results. The error to cost ratio is analyzed in details for a few cases. The results suggest that high order schemes can be more computationally efficient than low order schemes.

On derivative estimation and the solution of least squares problems

Surface interpolation finds application in many aspects of science and technology. Two specific areas of interest are surface reconstruction techniques for plant architecture and approximating cell face fluxes in the finite volume discretisation strategy for solving partial differential equations numerically. An important requirement of both applications is accurate local gradient estimation. In surface reconstruction this gradient information is used to increase the accuracy of the local interpolant, while in the finite volume framework accurate gradient information is essential to ensure second order spatial accuracy of the discretisation. In this work two different least squares strategies for approximating these local gradients are investigated and the errors associated with each analysed. It is shown that although the two strategies appear different, they produce the same least squares error. Some carefully chosen case studies are used to elucidate this finding.

A sub-cell based indicator for troubled zones in RKDG schemes and a novel class of hybrid RKDG+HWENO schemes

Runge–Kutta Discontinuous Galerkin (RKDG) schemes can provide highly accurate solutions for a large class of important scientific problems. Using them for problems with shocks and other discontinuities requires that one has a strategy for detecting the presence of these discontinuities. Strategies that are based on total variation diminishing (TVD) limiters can be problem-independent and scale-free but they can indiscriminately clip extrema, resulting in degraded accuracy. Those based on total variation bounded (TVB) limiters are neither problem-independent nor scale-free. In order to get past these limitations we realize that the solution in RKDG schemes can carry meaningful sub-structure within a zone that may not need to be limited. To make this sub-structure visible, we take a sub-cell approach to detecting zones with discontinuities, known as troubled zones. A monotonicity preserving (MP) strategy is applied to distinguish between meaningful sub-structure and shocks. The strategy does not indiscriminately clip extrema and is, nevertheless, scale-free and problem-independent. It, therefore, overcomes some of the limitations of previously-used strategies for detecting troubled zones. The moments of the troubled zones can then be corrected using a weighted essentially non-oscillatory (WENO) or Hermite WENO (HWENO) approach. In the course of doing this work it was also realized that the most significant variation in the solution is contained in the solution variables and their first moments. Thus the additional moments can be reconstructed using the variables and their first moments, resulting in a very substantial savings in computer memory. We call such schemes hybrid RKDG+HWENO schemes. It is shown that such schemes can attain the same formal accuracy as RKDG schemes, making them attractive, low-storage alternatives to RKDG schemes. Particular attention has been paid to the reconstruction of cross-terms in multi-dimensional problems and explicit, easy to implement formulae have been catalogued for third and fourth order of spatial accuracy. The utility of hybrid RKDG+WENO schemes has been illustrated with several stringent test problems in one and two dimensions. It is shown that their accuracy is usually competitive with the accuracy of RKDG schemes of the same order. Because of their compact stencils and low storage, hybrid RKDG+HWENO schemes could be very useful for large-scale parallel adaptive mesh refinement calculations.

Energy relaxation method for chemical non‐equilibrium flow computations

AbstractIn this paper, an algorithm for chemical non‐equilibrium hypersonic flow is developed based on the concept of energy relaxation method (ERM). The new system of equations obtained are studied using finite volume method with Harten–Lax–van Leer scheme for contact (HLLC). The original HLLC method is modified here to account for additional species and split energy equations. Higher order spatial accuracy is achieved using MUSCL reconstruction of the flow variables with van Albada limiter. The thermal equilibrium is considered for the analysis and the species data are generated using polynomial correlations. The single temperature model of Dunn and Kang is used for chemical relaxation. The computed results for a flow field over a hemispherical cylinder at Mach number of 16.34 obtained using the present solver are found to be promising and computationally (25%) more efficient. The present solver captures physically correct solution as the entropy conditions are satisfied automatically during the computations. Copyright © 2007 John Wiley & Sons, Ltd.

Order of accuracy analysis for multiresolution time‐domain using Daubechies bases

AbstractIn this paper, the spatial order of accuracy of multiresolution time‐domain methods using basis functions from the Daubechies family are studied. It is observed that MRTD methods using scaling functions from the Daubechies N‐vanishing‐moment orthonormal family have a spatial order of accuracy equal to 2N. For MRTD methods using scaling functions from the CDF (2, N2) biorthogonal family, the spatial order of accuracy is 2 + N2. We also find that the order of accuracy is the same for both staggered and nonstaggered spatial grids. The simulation results are presented which confirm analytical predictions. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 45: 290–293, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20798

On the admissibility of stencils for first order polynomial reconstruction on three-dimensional unstructured meshes

Starting from the work of Abgrall [1], several authors proposed finite volume methods for the solution of fluid dynamics problems on unstructured meshes, with order of spatial accuracy greater than two and possessing non-oscillatory properties. A recent review of these methods can be found in [2]. Also Sonar [3] gives formal proofs of several properties of non-oscillatory schemes on unstructured meshes. A technique that allows obtaining spatial order of accuracy three or four was developed by Hu and Shu [4]. On this weighted essentially non-oscillatory scheme, first order-polynomials are used to obtain reconstructions of higher order spatial accuracy. Thus, admissibility of the corresponding stencils is an important question to be addressed in considering possible extensions to the 3D case. In two dimensions, the equivalence between admissibility of a three control volume stencil and collinearity of the mass centers is a known fact [5]. In this work we prove that admissibility of a three-dimensional four control volumes stencil is equivalent to the fact of the mass centers not being coplanar. The technique of proof can easily be adapted to the two-dimensional case for the analogous result as well as for proving admissibility in higher order spaces without difficulties.

Use of slopelimiter techniques in traditional numerical methods for multi-phase flow in pipelines and wells

The aim of this paper is to show how simple and traditional methods for simulating multi-phase flow can be improved by introducing higher order accuracy. Numerical diffusion is reduced to a minimum by using slopelimiter techniques, and better predictions of flow rates and pressure are obtained. Slopelimiter techniques, originally developed to achieve higher order of accuracy in Godunov's method, is applied to a method following a finite element approach and a predictor–corrector shooting technique. These methods are tested and compared to a Godunov-type scheme recently developed for multi-phase flow. Implementation of Godunov-type schemes for multi-phase flow tends to be a complicated and challenging task. Introducing the slopelimiter techniques in the finite element approach and the predictor–corrector shooting technique is however simple, and provides an overall reduction of the numerical diffusion. The focus is on using these techniques to improve the mass transport description, since this is the main concern in the applications needed. The presented schemes represent different semi-implicit approaches for simulating multi-phase flow. An evaluation of higher order extensions, as well as a comparison by itself, is of large interest. We present a model for two-phase flow in pipelines and wells, and an outline of the numerical methods and the extensions to second order spatial accuracy. Several examples motivated by applications in underbalanced drilling are presented, and the advantages of using higher order schemes are illustrated. Copyright © 2005 John Wiley & Sons, Ltd.

Fourth-order compact formulation of Navier-Stokes equations and driven cavity flow at high Reynolds numbers

A new fourth order compact formulation for the steady 2-D incompressible Navier-Stokes equations is presented. The formulation is in the same form of the Navier-Stokes equations such that any numerical method that solve the Navier-Stokes equations can also be applied to this fourth order compact formulation. In particular in this work the formulation is solved with an efficient numerical method that requires the solution of tridiagonal systems using a fine grid mesh of 601x601. Using this formulation, the steady 2-D incompressible flow in a driven cavity is solved up to Reynolds number of 20,000 with fourth order spatial accuracy. Detailed solutions are presented.

Verification of Euler/Navier–Stokes codes using the method of manufactured solutions

AbstractThe method of manufactured solutions is used to verify the order of accuracy of two finite‐volume Euler and Navier–Stokes codes. The Premo code employs a node‐centred approach using unstructured meshes, while the Wind code employs a similar scheme on structured meshes. Both codes use Roe's upwind method with MUSCL extrapolation for the convective terms and central differences for the diffusion terms, thus yielding a numerical scheme that is formally second‐order accurate. The method of manufactured solutions is employed to generate exact solutions to the governing Euler and Navier–Stokes equations in two dimensions along with additional source terms. These exact solutions are then used to accurately evaluate the discretization error in the numerical solutions. Through global discretization error analyses, the spatial order of accuracy is observed to be second order for both codes, thus giving a high degree of confidence that the two codes are free from coding mistakes in the options exercised. Examples of coding mistakes discovered using the method are also given. Copyright © 2004 John Wiley & Sons, Ltd.

Difference Approximations of the Neumann Problem for the Second Order Wave Equation

Stability theory and numerical experiments are presented for a finite difference method that directly discretizes the Neumann problem for the second order wave equation. Complex geometries are discretized using a Cartesian embedded boundary technique. Both second and third order accurate approximations of the boundary conditions are presented. Away from the boundary, the basic second order method can be corrected to achieve fourth order spatial accuracy. To integrate in time, we present both a second order and a fourth order accurate explicit method. The stability of the method is ensured by adding a small fourth order dissipation operator, locally modified near the boundary to allow its application at all grid points inside the computational domain. Numerical experiments demonstrate the accuracy and long-time stability of the proposed method.

Finite difference method of simulation of non-steady-state ion transfer in electrochemical systems with allowance for migration

Finite difference methods of the second order of accuracy are elaborated for numerical calculation of non-steady-state ion transfer, which is caused by diffusion, migration, and convection in the unidimensional electrochemical systems. The methods of decoupling a set of coupled continuity equations of the electrolyte species are proposed, which ensures that the discrete equations are consistent with the initial differential equations and the electroneutrality condition is rigorously met. The methods of approximation of the boundary conditions of the second order temporal and spatial accuracy and the method of decoupling the transfer equations in the boundary nodes are elaborated. The explicit, fully implicit, and semi-implicit finite difference schemes are elaborated. For semi-implicit schemes, two versions of difference equation closure are proposed, which assure the unambiguity of determination of the distribution of electrical potential. Comparison analysis of the accuracy of elaborated finite difference methods of calculation of non-steady-state ion transfer is performed.

An adaptive solution of the 3-D Euler equations on an unstructured grid

A new algorithm to generate the unstructured grid on a curved surface is developed. The advancing front method is used to generate the tetrahedral meshes in the space. An adaptive grid technique is used to enhance the calculation efficiency. The AUSM+ (Advection Upstream Splitting Method) scheme which was developed on a structured grid has been extended to be used to the spatial discretization of a cell-centered finite volume formulation on the unstructured grid. A second order spatial accuracy is achieved by applying a novel cell reconstruction procedure which can prevent the solution from exhibiting spurious oscillations without adding a limiter. A 3-D Euler solver for an adaptive tetrahedral grid and numerical results for several cases are presented.