Order Accuracy In Space Research Articles

Parallel implementation of a velocity-stress staggered-grid finite-difference method for 2-D poroelastic wave propagation

Numerical simulation of wave propagation in poroelastic media demands significantly more computational capability compared to elastic media simulation. Use of serial codes in a single scientific workstation limits the size of problem. To overcome this difficulty, a parallel velocity-stress staggered-grid finite-difference method is developed for efficient simulation of wave propagation in 2-D poroelastic media. The finite difference formulation of Biot's theory has the properties of fourth order accuracy in space and second order accuracy in time. The model is decomposed into small subdomains for each processor. After each processor updates wavefields within its domain, the processors exchange the wavefields via message passing interface (MPI). The parallel implementation reduces the computational time and also allows one to study larger problems. From our numerical experiment, consistent with other 1-D experiments, it is found that the presence of heterogeneity of porous medium can produce significant P-wave attenuation in the seismic frequency range.

ADER discontinuous Galerkin schemes for aeroacoustics

In this paper we apply the ADER approach to the Discontinuous Galerkin (DG) framework for the two-dimensional linearized Euler equations. The result is an efficient high order accurate single-step scheme in time which uses less storage than Runge–Kutta DG schemes, especially for very high order of accuracy. The aim is to obtain an arbitrarily accurate scheme in space and time on unstructured grids for accurate noise propagation in the time domain in very complex geometries. We will present numerical convergence rates for ADER-DG methods up to 10th order of accuracy in space and time on structured and unstructured meshes. To cite this article: M. Dumbser, C.-D. Munz, C. R. Mecanique 333 (2005).

A numerical scheme for the integration of the Vlasov–Poisson system of equations, in the magnetized case

We present a numerical algorithm for the solution of the Vlasov–Poisson system of equations, in the magnetized case. The numerical integration is performed using the well-known “splitting” method in the electrostatic approximation, coupled with a finite difference upwind scheme; finally the algorithm provides second order accuracy in space and time. The cylindrical geometry is used in the velocity space, in order to describe the rotation of the particles around the direction of the external uniform magnetic field. Using polar coordinates, the integration of the Vlasov equation is very simplified in the velocity space with respect to the cartesian geometry, because the rotation in the velocity cartesian space corresponds to a translation along the azimuthal angle in the cylindrical reference frame. The scheme is intrinsically symplectic and significatively simpler to implement, with respect to a cartesian one. The numerical integration is shown in detail and several conservation tests are presented, in order to control the numerical accuracy of the code and the time evolution of the entropy, strictly related to the filamentation problem for a kinetic model, is discussed.

High order Godunov mixed methods on tetrahedral meshes for density driven flow simulations in porous media

Two-dimensional Godunov mixed methods have been shown to be effective for the numerical solution of density-dependent flow and transport problems in groundwater even when concentration gradients are high and the process is dominated by density effects. This class of discretization approaches solves the flow equation by means of the mixed finite element method, thus guaranteeing mass conserving velocity fields, and discretizes the transport equation by mixed finite element and finite volumes techniques combined together via appropriate time splitting. In this paper, we extend this approach to three dimensions employing tetrahedral meshes and introduce a spatially variable time stepping procedure that improves computational efficiency while preserving accuracy by adapting the time step size according to the local Courant–Friedrichs–Lewy (CFL) constraint. Careful attention is devoted to the choice of a truly three-dimensional limiter for the advection equation in the time-splitting technique, so that to preserve second order accuracy in space (in the sense that linear functions are exactly interpolated). The three-dimensional Elder problem and the saltpool problem, recently introduced as a new benchmark for testing three-dimensional density models, provide assessments with respect to accuracy and reliability of this numerical approach.

A HIGH ORDER KINETIC FLUX-SPLITTING METHOD FOR THE SPECIAL RELATIVISTIC HYDRODYNAMICS

In this article we present a flux splitting method based on gas-kinetic theory for the special relativistic hydrodynamics (SRHD) [Landau and Lifshitz, Fluid Mechanics, Pergamon New York, 1987] in one and two space dimensions. This kinetic method is based on the direct splitting of the macroscopic flux functions with the consideration of particle transport. At the same time, particle "collisions" are implemented in the free transport process to reduce numerical dissipation. Due to the nonlinear relations between conservative and primitive variables and the consequent complexity of the Jacobian matrix, the multi-dimensional shock-capturing numerical schemes for SRHD are computationally more expensive. All the previous methods presented for the solution of these equations were based on the macroscopic continuum description. These upwind high-resolution shock-capturing (HRSC) schemes, which were originally made for non-relativistic flows, were extended to SRHD. However our method, which is based on kinetic theory is more related to the physics of these equations and is very efficient, robust, and easy to implement. In order to get high order accuracy in space, we use a third order central weighted essentially non-oscillatory (CWENO) finite difference interpolation routine. To achieve high order accuracy in time we use a Runge-Kutta time stepping method. The one- and two-dimensional computations reported in this paper show the desired accuracy, high resolution, and robustness of the method.

A wind driven three-dimensional pollutant transport model

This paper describes a computer program that can be used to simulate three-dimensional wind driven pollutant transport under different environmental conditions. The model uses the three-dimensional governing equations for wind-induced circulation in lakes as determined from the Reynolds stress equations. The governing equations are approximated by the control volume method on the Arakawa-C staggered grid. The model accounts for inflows and outflows from major tributaries and the Coriolis effect. A semi-implicit difference scheme is used for the barotropic pressure, the Adams–Bashford scheme for the temporal terms and the weight averaged Donor-cell scheme for the advective terms. The central difference scheme, of second order accuracy in space, is used for the diffusive term. The functionality of the model is demonstrated using two simple test basins under a single source and constant wind conditions. In addition the applicability of the model to a nearshore, multiple source, variable wind case is illustrated.

A Two-Dimensional Nonlinear Nonlocal Feed-Forward Cochlear Model and Time Domain Computation of Multitone Interactions

A two-dimensional cochlear model is presented, which couples the classical second order partial differential equations of the basilar membrane (BM) with a discrete feed-forward (FF) outer hair cell (OHC) model for enhanced sensitivity. The enhancement (gain) factor in the model depends on BM displacement in a nonlinear nonlocal manner in order to capture multifrequency sound interactions and compression effects in a time-dependent simulation. The FF mechanism is based on the longitudinal tilt of the OHCs in feeding the mechanical energy onto the BM. A numerical method of second order accuracy in space and time is formulated by reducing the unknown variables to the BM with the representation of eigenfunction expansions. Though the nonlinear coupling with OHCs created an implicit algebraic problem at each time step, the structure of the FF mass matrix is found to permit a decomposition into a sum of a time-independent symmetric positive definite part and the remaining time-dependent part. A fast iterative m...

A high order mixed vector finite element method for solving the time dependent Maxwell equations on unstructured grids

We present a mixed vector finite element method for solving the time dependent coupled Ampere and Faraday laws of Maxwell’s equations on unstructured hexahedral grids that employs high order discretization in both space and time. The method is of arbitrary order accuracy in space and up to 4th order accurate in time, making it well suited for electrically large problems where grid anisotropy and numerical dispersion have plagued other methods. In addition, the method correctly models both the jump discontinuities and the divergence-free properties of the electric and magnetic fields, is charge and energy conserving, conditionally stable, and free of spurious modes. Several computational experiments are performed to demonstrate the accuracy, efficiency and benefits of the method.

Smagorinsky model을 이용한 실린더 및 익형 주위의 LES 난류유동해석

As a computer has been continuously progressed to reduce R&D time and cost, the study of the flow physics has been significantly relied on the numerical method. Recently, Large Eddy Simulation(LES) has been widely used in CFD community to accurately capture the turbulent flows. The LES code requires high accuracy in time, as well as in space. Also, it should have strong robustness to ensure the convergence in various complicated flows. The objective of the present work is to develop a base code for LES simulation, having 2<TEX>$^{nd}$</TEX> order accuracy in time and 4<TEX>$^{th}$</TEX> order accuracy in space. To achieve the present objective, the four-step fractional step method was enhanced by adopting compact Pade'scheme. The standard Smagorinsky model was implemented for the first stage of the present code development. The flows over a cylinder and an airfoil were successfully simulated. and an airfoil were successfully simulated.

A Generalized Relaxation Method for Transport and Diffusion of Pollutant Models in Shallow Water

AbstractWe present a numerical method based on finite difference relaxation approximations for computing the transport and diffusion of a passive pollutant by a water flow. The flow is modeled by the well-known shallow water equations and the pollutant propagation is described by a transport equation. The previously developed nonoscillatory relaxation scheme is generalized to cover problems with pollutant trans- port, in one and two dimensions and source terms, resulting in a class of methods of the first and the second order of accuracy in space and time. The methods are based on the classical relaxation models combined with a Runge-Kutta time splitting scheme, where neither Riemann solvers nor characteristic decompositions are needed. Numerical results are presented for several benchmark test problems. The schemes presented are verified by comparing the results with documented ones, proving that no special treatment is needed for the transport equation in order to obtain accurate results.

A Free, Fast, Simple, and Efficient Total Variation Diminishing Magnetohydrodynamic Code

We describe a numerical method to solve the magnetohydrodynamic (MHD) equations. The fluid variables are updated along each direction using the flux conservative, 2nd order, total variation diminishing (TVD), upwind scheme of Jin and Xin. The magnetic field is updated separately in two-dimensional advection-constraint steps. The electromotive force (EMF) is computed in the advection step using the TVD scheme, and this same EMF is used immediately in the constraint step in order to preserve \grad.B=0 without the need to store intermediate fluxes. Operator splitting is used to extend the code to three dimensions, and Runge-Kutta is used to get second order accuracy in time. The advantages of this code are high resolution per grid cell, second order accuracy in space and time, enforcement of the \grad.B=0 constraint to machine precision, no memory overhead, speed, and simplicity. A 3-D Fortran implementation less than 400 lines long is made freely available. We also implemented a fully scalable message-passing parallel MPI version. We present tests of the code on MHD waves and shocks.

Numerical methods for the generalized Zakharov system

We present two numerical methods for the approximation of the generalized Zakharov system (ZS). The first one is the time-splitting spectral (TSSP) method, which is explicit, time reversible, and time transverse invariant if the generalized ZS is, keeps the same decay rate of the wave energy as that in the generalized ZS, gives exact results for the plane-wave solution, and is of spectral-order accuracy in space and second-order accuracy in time. The second one is to use a local spectral method, the discrete singular convolution (DSC) for spatial derivatives and the fourth-order Runge–Kutta (RK4) for time integration, which is of high (the same as spectral)-order accuracy in space and can be applied to deal with general boundary conditions. In order to test accuracy and stability, we compare these two methods with other existing methods: Fourier pseudospectral method (FPS) and wavelet-Galerkin method (WG) for spatial derivatives combining with the RK4 for time integration, as well as the standard finite difference method (FD) for solving the ZS with a solitary-wave solution. Furthermore, extensive numerical tests are presented for plane waves, solitary-wave collisions in 1d, as well as a 2d problem of the generalized ZS. Numerical results show that TSSP and DSC are spectral-order accuracy in space and much more accurate than FD, and for stability, TSSP requires k=O( h), DSC–RK4 requires k=O( h 2) for fixed acoustic speed, where k is the time step and h is the spatial mesh size.

High order numerical methods for the space non-homogeneous Boltzmann equation

In this paper we present accurate methods for the numerical solution of the Boltzmann equation of rarefied gas. The methods are based on a time splitting technique. The transport is solved by a third order accurate (in space) positive and flux conservative (PFC) method. The collision step is treated by a Fourier approximation of the collision integral, which guarantees spectral accuracy in velocity, coupled with several high order integrators in time. Strang splitting is used to achieve second order accuracy in space and time. Several numerical tests illustrate the properties of the methods.

Finite volume method for simulating extreme flood events in natural channels

The need for mitigating damages produced by extreme hydrologie events has stimulated the European Community to fund several projects. The Concerted Action on Dam-break Modelling workgroup (CADAM) performed a considerable work for the development of new codes and for the adequate verification of their performance. In the context of the CADAM project, a new 2D computer code is developed, tested and applied, as described in the present paper. The algorithm is obtained through the spatial discretisation of the shallow water equations by a finite volume method, based on the Godunov approach. The HLL Riemann solver is used. A second order accuracy in space and time is achieved, respectively by MUSCL and predictor-corrector techniques. The high resolution requirement is ensured by satisfaction of TVD property. Particular attention is posed to the numerical treatment of source terms. Accuracy, stability and the reliability of the code are tested on a selected set of study cases. A grid refinement analysis is performed. Numerical results are compared with experimental data, obtained by the physical modelling of a submersion wave on a portion of the Toce river valley, Italy, performed by ENEL-HYDRO and considered as representative of a real life flood occurrence.

TECHNIQUE FOR VERY HIGH ORDER NONLINEAR SIMULATION AND VALIDATION

Finding the sources of sound in large nonlinear fields via direct simulation currently requires excessive computational cost. This paper describes a simple technique for efficiently solving the multidimensional nonlinear Euler equations that significantly reduces this cost and demonstrates a useful approach for validating high order nonlinear methods. Up to 15th order accuracy in space and time methods were compared and it is shown that an algorithm with a fixed design accuracy approaches its maximal utility and then its usefulness exponentially decays unless higher accuracy is used. It is concluded that at least a 7th order method is required to efficiently propagate a harmonic wave using the nonlinear Euler equations to a distance of five wavelengths while maintaining an overall error tolerance that is low enough to capture both the mean flow and the acoustics.

Technique for very High Order Nonlinear Simulation and Validation

Finding the sources of sound in large nonlinear fields via direct simulation currently requires excessive computational cost. This paper describes a simple technique for efficiently solving the multidimensional nonlinear Euler equations that significantly reduces this cost and demonstrates a useful approach for validating high order nonlinear methods. Up to 15th order accuracy in space and time methods were compared and it is shown that an algorithm with a fixed design accuracy approaches its maximal utility and then its usefulness exponentially decays unless higher accuracy is used. It is concluded that at least a 7th order method is required to efficiently propagate a harmonic wave using the nonlinear Euler equations to a distance of 5 wavelengths while maintaining an overall error tolerance that is low enough to capture both the mean flow and the acoustics.

ADER: Arbitrary High Order Godunov Approach

This paper concerns the construction of non-oscillatory schemes of very high order of accuracy in space and time, to solve non-linear hyperbolic conservation laws. The schemes result from extending the ADER approach, which is related to the ENO/WENO methodology. Our schemes are conservative, one-step, explicit and fully discrete, requiring only the computation of the inter-cell fluxes to advance the solution by a full time step; the schemes have optimal stability condition. To compute the intercell flux in one space dimension we solve a generalised Riemann problem by reducing it to the solution a sequence of m conventional Riemann problems for the kth spatial derivatives of the solution, with ke0, 1,m, m−1, where m is arbitrary and is the order of the accuracy of the resulting scheme. We provide numerical examples using schemes of up to fifth order of accuracy in both time and space.

Discrete-Time Orthogonal Spline Collocation Methods for Vibration Problems

Discrete-time orthogonal spline collocation schemes are formulated and analyzed for vibration problems involving various boundary conditions. Each problem is written as a Schrödinger-type system, which is then approximated by Crank--Nicolson and/or alternating direction implicit orthogonal spline collocation schemes. These schemes are shown to be second-order accurate in time and of optimal order accuracy in space in the Hm-norm, m=1,2.

Use of an upwind finite volume method to solve the air bearing problem of hard disk drives

In this paper we present an upwind higher order finite volume numerical scheme over unstructured triangular mesh to solve the slider air bearing problem of hard disk drives. The scheme is nodal based, which uses the median dual as the control volume. The convection part of the generalized Reynolds equation is modeled by the flux difference splitting technique. Higher order accuracy in space is achieved by a linear reconstruction technique with a flux limiting technique incorporated to prevent oscillation in the high-pressure gradient regions. A linear Galerkin method is used to discretize the diffusion terms. In addition, a non-nested multi-grid iteration technique is used to increase the convergence rate. Finally, the steady state flying attitude of a slider subject to pre-applied suspension force and torques is obtained by a Quasi–Newton iteration method, and the results of this scheme are compared with two other schemes.

A Crank–Nicolson orthogonal spline collocation method for vibration problems

A discrete-time orthogonal spline collocation scheme is formulated and analyzed for a problem governing the transverse vibrations of a clamped square plate. The problem is reformulated as a Schrödinger-type system which is then approximated by a Crank–Nicolson orthogonal spline collocation scheme. This scheme is shown to be second-order accurate in time and of optimal order accuracy in space in the H 1 - and H 2 -norms.