Showing papers in "Journal of Scientific Computing in 2004"
TL;DR: This paper presents dissipation operators that preserve both stability and accuracy for high order finite difference approximations of initial boundary value problems.
Abstract: Stability for nonlinear convection problems using centered difference schemes require the addition of artificial dissipation. In this paper we present dissipation operators that preserve both stability and accuracy for high order finite difference approximations of initial boundary value problems.
189 citations
TL;DR: A major focus of this paper is to verify that the SV method is capable of achieving high-order accuracy for hyperbolic systems of conservation laws.
Abstract: In this paper, the third in a series, the Spectral Volume (SV) method is extended to one-dimensional systems—the quasi-1D Euler equations. In addition, several new partitions are identified which optimize a certain form of the Lebesgue constant, and the performance of these partitions is assessed with the linear wave equation. A major focus of this paper is to verify that the SV method is capable of achieving high-order accuracy for hyperbolic systems of conservation laws. Both steady state and time accurate problems are used to demonstrate the overall capability of the SV method.
163 citations
TL;DR: It is shown that it is impossible to construct a coordinate transformation operator for high order finite difference methods obeying a summation-by-parts rule without decreasing the order of accuracy of the method.
Abstract: High order finite difference methods obeying a summation-by-parts (SBP) rule are developed for equidistant grids. With curvilinear grids, a coordinate transformation operator that does not destroy the SBP property must be used. We show that it is impossible to construct such an operator without decreasing the order of accuracy of the method.
113 citations
TL;DR: Monotonicity issues for arbitrary norms and linear and nonlinear problems are considered and some known results are related with the ones obtained in the SSP context.
Abstract: Over the last few years, great effort has been made to develop high order strong stability preserving (SSP) Runge–Kutta methods. These methods have a nonlinear stability property that makes them suitable for the time integration of ODEs that arise from a method of lines approximation of hyperbolic conservation laws. Basically, this stability property is a monotonicity property for the internal stages and the numerical solution. Recently Ferracina and Spijker have established a link between stepsize restrictions for monotonicity and the already known stepsize restrictions for contractivity. Hence the extensive research on contractivity can be transferred to the SSP context. In this paper we consider monotonicity issues for arbitrary norms and linear and nonlinear problems. We collect and review some known results and relate them with the ones obtained in the SSP context.
106 citations
TL;DR: To minimize the tuning of parameters and physical problem dependence, new sensors with improved detection properties are proposed, derived from utilizing appropriate non-orthogonal wavelet basis functions and they can be used to completely switch off the extra numerical dissipation outside shock layers.
Abstract: The recently developed essentially fourth-order or higher low dissipative shock-capturing scheme of Yee, Sandham, and Djomehri l25r aimed at minimizing numerical dissipations for high speed compressible viscous flows containing shocks, shears and turbulence. To detect non-smooth behavior and control the amount of numerical dissipation to be added, Yee et al. employed an artificial compression method (ACM) of Harten l4r but utilize it in an entirely different context than Harten originally intended. The ACM sensor consists of two tuning parameters and is highly physical problem dependent. To minimize the tuning of parameters and physical problem dependence, new sensors with improved detection properties are proposed. The new sensors are derived from utilizing appropriate non-orthogonal wavelet basis functions and they can be used to completely switch off the extra numerical dissipation outside shock layers. The non-dissipative spatial base scheme of arbitrarily high order of accuracy can be maintained without compromising its stability at all parts of the domain where the solution is smooth. Two types of redundant non-orthogonal wavelet basis functions are considered. One is the B-spline wavelet (Mallat and Zhong l14r) used by Gerritsen and Olsson l3r in an adaptive mesh refinement method, to determine regions where refinement should be done. The other is the modification of the multiresolution method of Harten l5r by converting it to a new, redundant, non-orthogonal wavelet. The wavelet sensor is then obtained by computing the estimated Lipschitz exponent of a chosen physical quantity (or vector) to be sensed on a chosen wavelet basis function. Both wavelet sensors can be viewed as dual purpose adaptive methods leading to dynamic numerical dissipation control and improved grid adaptation indicators. Consequently, they are useful not only for shock-turbulence computations but also for computational aeroacoustics and numerical combustion. In addition, these sensors are scheme independent and can be stand-alone options for numerical algorithms other than the Yee et al. scheme.
104 citations
TL;DR: This work presents a numerical implementation of the fast Galerkin method for Fredholm integral equations of the second kind using the piecewise polynomial wavelets and implements a multiscale iteration method for solving the resulting compressed linear system.
Abstract: We present a numerical implementation of the fast Galerkin method for Fredholm integral equations of the second kind using the piecewise polynomial wavelets. We focus on addressing critical issues for the numerical implementation of such a method. They include a choice of practical truncation strategy, numerical integration of weakly singular integrals and the error control of the numerical quadrature. We also implement a multiscale iteration method for solving the resulting compressed linear system. Numerical examples are given to demonstrate the proposed ideas and methods.
61 citations
TL;DR: 3-D large eddy simulation (LES) results for a turbulent Mach 0.9 isothermal round jet at a Reynolds number of 100,000 are presented and comparisons are made with experimental measurements of jets at similar flow conditions.
Abstract: We present 3-D large eddy simulation (LES) results for a turbulent Mach 0.9 isothermal round jet at a Reynolds number of 100,000 (based on jet nozzle exit conditions and nozzle diameter). Our LES code is part of a Computational Aeroacoustics (CAA) methodology that couples surface integral acoustics techniques such as Kirchhoff's method and the Ffowcs Williams-- Hawkings method with LES for the far field noise estimation of turbulent jets. The LES code employs high-order accurate compact differencing together with implicit spatial filtering and state-of-the-art non-reflecting boundary conditions. A localized dynamic Smagorinsky subgrid-scale (SGS) model is used for representing the effects of the unresolved scales on the resolved scales. A computational grid consisting of 12 million points was used in the present simulation. Mean flow results obtained in our simulation are found to be in very good agreement with the available experimental data of jets at similar flow conditions. Furthermore, the near field data provided by the LES is coupled with the Ffowcs Williams--Hawkings method to compute the far field noise. Far field aeroacoustics results are also presented and comparisons are made with experimental measurements of jets at similar flow conditions. The aeroacoustics results are encouraging and suggest further investigation of the effects of inflow conditions on the jet acoustic field.
57 citations
TL;DR: A very simple but an effective method is proposed here in early diagnosis for evanescent discontinuities using a third order one-sided stencil at the discontinuity to produce solution with vastly reduced Gibbs' phenomenon of the solution.
Abstract: Compact difference schemes have been investigated for their ability to capture discontinuities. A new proposed scheme (Sengupta, Ganerwal and De (2003). J. Comp. Phys. 192(2), 677.) is compared with another from the literature Zhong (1998). J. Comp. Phys. 144, 622 that was developed for hypersonic transitional flows for their property related to spectral resolution and numerical stability. Solution of the linear convection equation is obtained that requires capturing discontinuities. We have also studied the performance of the new scheme in capturing discontinuous solution for the Burgers equation. A very simple but an effective method is proposed here in early diagnosis for evanescent discontinuities. At the discontinuity, we switch to a third order one-sided stencil, thereby retaining the high accuracy of solution. This produces solution with vastly reduced Gibbs' phenomenon of the solution. The essential causes behind Gibbs' phenomenon is also explained.
46 citations
Journal Article•
TL;DR: An effective method is given for computing the entropy for polynomials orthogonal on a segment of the real axis, which uses as input data only the coefficients of the recurrence relation satisfied by these polynmials.
Abstract: We give an effective method to compute the entropy for polynomials orthogonal on a segment of the real axis that uses as input data only the coefficients of the recurrence relation satisfied by these polynomials. This algorithm is based on a series expression for the mutual energy of two probability measures naturally connected with the polynomials. The particular case of Gegenbauer polynomials is analyzed in detail. These results are applied also to the computation of the entropy of spherical harmonics, important for the study of the entropic uncertainty relations as well as the spatial complexity of physical systems in central potentials.
46 citations
TL;DR: This system appears to be a very interesting test problem for any anti dissipative scheme for conservation laws and a numerical scheme in which the numerical dissipation is controlled in such a way that the results for large time are meaningful is proposed.
Abstract: We numerically investigate the long time behavior of solutions of the Lifshitz– Slyozov system. We propose a numerical scheme in which the numerical dissipation is controlled in such a way that the results for large time are meaningful. In this respect, we find the long time behavior to crucially depend on the distribution of largest aggregates present in the solution. This fact proved, in some particular cases in (23), was difficult to obtain with previous numerical schemes in the engineering literature leading to wrong statements. We propose a numerical scheme in which we can observe and quantify the equilibration rates towards the right asymptotic profile. Moreover, this system appears to be a very interesting test problem for any anti dissipative scheme for conservation laws.
44 citations
TL;DR: The benefits of the adaptive grid refinement process are illustrated by numerical results that show how the method works and its efficiency if compared to the classical conforming Finite Element Method.
Abstract: This paper deals with the efficient and accurate computation of extracellular potentials in a simplified model of myocardial tissue. The electrical activity of the heart is characterized by a narrow wavefront spreading through the myocardium. To increase the accuracy of the computation, a non-conforming non-overlapping domain decomposition based on the mortar method is used, allowing adaptivity in the regions closed to the wavefront. The benefits of the adaptive grid refinement process are illustrated by numerical results that show how the method works and its efficiency if compared to the classical conforming Finite Element Method.
TL;DR: It is shown that the presence of the computational mode in Adams–Bashforth scheme leads to erroneous results, if the solution contains high frequency components, and shows that second order Adams– Bashforth time integration is not suitable for DNS.
Abstract: A qualitative and quantitative study is made for choosing time advancement strategies for solving time dependent equations accurately. A single step, low order Euler time integration method is compared with Adams–Bashforth, a second order accurate time integration strategy for the solution of one dimensional wave equation. With the help of the exact solution, it is shown that the presence of the computational mode in Adams–Bashforth scheme leads to erroneous results, if the solution contains high frequency components. This is tested for the solution of incompressible Navier–Stokes equation for uniform flow past a rapidly rotating circular cylinder. This flow suffers intermittent temporal instabilities implying presence of high frequencies. Such instabilities have been noted earlier in experiments and high accuracy computations for similar flow parameters. This test problem shows that second order Adams– Bashforth time integration is not suitable for DNS.
TL;DR: This work introduces a new class of schemes for monotone scalar conservation laws that satisfy an entropy inequality, while still resolving exactly the single traveling shocks or contact discontinuities, and shows that it is then possible to have an excellent resolution of rarefaction waves, and also to avoid the undesirable staircase effect.
Abstract: In a recent work J. Sci. Comput. 16, 479–524 (2001), B. Despres and F. Lagoutiere introduced a new approach to derive numerical schemes for hyperbolic conservation laws. Its most important feature is the ability to perform an exact resolution for a single traveling discontinuity. However their scheme is not entropy satisfying and can keep nonentropic discontinuities. The purpose of our work is, starting from the previous one, to introduce a new class of schemes for monotone scalar conservation laws, that satisfy an entropy inequality, while still resolving exactly the single traveling shocks or contact discontinuities. We show that it is then possible to have an excellent resolution of rarefaction waves, and also to avoid the undesirable staircase effect. In practice, our numerical experiments show second-order accuracy.
TL;DR: This paper addresses both parameter optimization and reduction of the round off error for the Gegenbauer reconstruction method, and constructs a viable “black box” method for choosing parameters that guarantee both theoretical and numerical convergence, even at the jump discontinuities.
Abstract: The Gegenbauer reconstruction method has been successfully implemented to reconstruct piecewise smooth functions by both reducing the effects of the Gibbs phenomenon and maintaining high resolution in its approximation. However, it has been noticed in some applications that the method fails to converge. This paper shows that the lack of convergence results from both poor choices of the parameters associated with the method, as well as numerical round off error. The Gegenbauer polynomials can have very large amplitudes, particularly near the endpoints xe±1, and hence the approximation requires that the corresponding computed Gegenbauer coefficients be extremely small to obtain spectral convergence. As is demonstrated here, numerical round off error interferes with the ability of the computed coefficients to decay properly, and hence affects the method's overall convergence. This paper addresses both parameter optimization and reduction of the round off error for the Gegenbauer reconstruction method, and constructs a viable “black box” method for choosing parameters that guarantee both theoretical and numerical convergence, even at the jump discontinuities. Validation of the Gegenbauer reconstruction method through a-posteriori estimates is also provided.
TL;DR: A parallel, unstructured, high-order discontinuous Galerkin method is developed for the time-dependent Maxwell's equations, using simple monomial polynomials for spatial discretization and a fourth-order Runge–Kutta scheme for time marching.
Abstract: A parallel, unstructured, high-order discontinuous Galerkin method is developed for the time-dependent Maxwell's equations, using simple monomial polynomials for spatial discretization and a fourth-order Runge–Kutta scheme for time marching. Scattering results for a number of validation cases are computed employing polynomials of up to third order. Accurate solutions are obtained on coarse meshes and grid convergence is achieved, demonstrating the capabilities of the scheme for time-domain electromagnetic wave scattering simulations.
TL;DR: A least-squares spectral collocation formulation for the Navier–Stokes problem is presented and the well known Babušska–Brezzi condition can be avoided.
Abstract: A least-squares spectral collocation formulation for the Navier–Stokes problem is presented. By this new approach the well known Babubska–Brezzi condition can be avoided. Here we are able to employ polynomials of the same degree both for the velocity components and for the pressure. The collocation conditions and the boundary conditions lead to a overdetermined system which can be efficiently solved by least-squares. The solution technique will only involve symmetric positive definite linear systems. The numerical simulations confirm the usual exponential rate of convergence for the spectral scheme.
TL;DR: Stability-like conditions are produced where periodic variations in semimajor and semiminor axes occur for extended periods of time, before orbital decay eventually takes over due to the effects of radiation reaction.
Abstract: The present study examines the behavior of a classical charged point particle in near-elliptic orbits about an infinitely massive and oppositely charged nucleus, while acted upon by applied electromagnetic radiation. As recently shown for near-circular orbits, and now extended here to the elliptical case, rather surprising nonlinear dynamical effects are readily produced for this simple system. A broad range of stability-like conditions can be achieved by applying radiation to this classical atom. A perfect balance condition is examined, which requires an infinite number of plane waves representing harmonics of the orbital motion. By applying a scale factor to this radiation, stability-like conditions are produced where periodic variations in semimajor and semiminor axes occur for extended periods of time, before orbital decay eventually takes over due to the effects of radiation reaction. This work is expected to lead to both practical suggestions on experimental ideas involving controlling ionization and stabilization conditions, as well as hopefully aiding in theoretical explorations of stochastic electrodynamics.
TL;DR: The Linearized Euler Equations (LEE) are solved on a uniform mesh for benchmark problems in one and two dimensions and a two dimensional mixing layer is solved by using Large-Eddy Simulation (LES).
Abstract: This paper presents the results of using high-order compact schemes with a high-order filter on multi-block domains. The Linearized Euler Equations (LEE) are solved on a uniform mesh for benchmark problems in one and two dimensions. Also a two dimensional mixing layer is solved by using Large-Eddy Simulation (LES). Three different boundary schemes are compared. The results compare well with the exact solutions and single-block domain results. The effect of the number of points of overlap among the subdomains is investigated. Having four points of overlap is chosen as a compromise between accuracy and efficiency.
IAC1
TL;DR: A numerical study of a buoyancy driven convection flow in presence of thermocapillarity, obtained very complex steady configurations for several values of the temperature difference at the lateral walls, ΔT=30, 40 and 50°C.
Abstract: A numerical study of a buoyancy driven convection flow in presence of thermocapillarity has been developed. The fluid is a silicone oil (Prandtl number equal to 105) contained in a three-dimensional box bounded by rigid and impermeable walls with top free surface exposed to a gaseous phase. At the lateral box walls a different non-uniform temperature distribution is assumed so to induce horizontal convection and to keep separated thermocapillary and buoyancy effects. The vorticity-velocity formulation of the time-dependent Navier–Stokes equations for a non-isothermal incompressible fluid is used. A procedure based on a linearized fully implicit finite difference second order scheme has been adopted. We obtained very complex steady configurations for several values of the temperature difference at the lateral walls, ΔTe30, 40 and 50°C. Along the direction perpendicular to the lateral walls, for ΔT increasing, we observe a physically meaningful growth of heat transfer. Confidence in these results is supported by a comparison with recent experimental and numerical observations.
TL;DR: A far wider range of phase conditions are found to provide stability than might intuitively be expected, with the time to orbital decay varying by orders of magnitude for any plane wave with an amplitude A above a critical value, Ac.
Abstract: The classical hydrogen atom is examined for the situation where a circularly polarized electromagnetic plane wave acts on a classical charged point particle in a near-circular orbit about an infinitely massive nucleus, with the plane wave normally incident to the plane of the orbit. The effect of the phase α of the polarized wave in relation to the velocity vector of the classical electron is examined in detail by carrying out a perturbation analysis and then comparing results using simulation methods. By expanding the variational parts of the radius and angular velocity about their average values, simpler nonlinear differential equations of motion are obtained that still retain the key features of the oscillating amplitude, namely, the gradual increase of the envelope of the oscillating amplitude and the point of rapid orbital decay. Also, as shown here, these key features carry over nicely to conventional quantities of interest such as energy and angular momentum. The phase α is shown here to have both subtle yet very significant effects on the quasistability of the orbital motion. A far wider range of phase conditions are found to provide stability than might intuitively be expected, with the time to orbital decay, td, varying by orders of magnitude for any plane wave with an amplitude A above a critical value, Ac.
TL;DR: Relaxed, essentially non-oscillating schemes for nonlinear conservation laws are presented, and it is possible to avoid the nonlinear Riemann problem, characteristic decompositions, and staggered grids.
Abstract: Relaxed, essentially non-oscillating schemes for nonlinear conservation laws are presented. Exploiting the relaxation approximation, it is possible to avoid the nonlinear Riemann problem, characteristic decompositions, and staggered grids. Nevertheless, convergence rates up to fourth order are observed numerically. Furthermore, a relaxed, piecewise hyperbolic scheme with artificial compression is constructed. Third order accuracy of this method is proved. Numerical results for two-dimensional Riemann problems in gas dynamics are presented. Finally, the relation to central schemes is discussed.
TL;DR: Two regularizations of an hyperbolic model system derived from the primitive equations of the ocean (or the atmosphere) are presented and the two regularized systems converge to different limits as the regularization parameter converges to 0.
Abstract: In this article two regularizations of an hyperbolic model system derived from the primitive equations of the ocean (or the atmosphere) are presented. The two regularized systems converge to different limits as the regularization parameter converges to 0. Numerical approximations of these equations and numerical simulations are also presented
TL;DR: A mathematical model treating of the dynamic contact line problem, supposed to describe the main features of the advancing triple line (rolling motion and variable contact angle) and to remove the singularity is studied.
Abstract: We study a mathematical model treating of the dynamic contact line problem, supposed to describe the main features of the advancing triple line (rolling motion and variable contact angle) and to remove the singularity. The model is composed by a macroscopic hydrodynamic free surface flow model (HFSM) (Navier--Stokes) coupled with a mesoscopic local surface model (LSM). Detailed mathematical and numerical analysis of the 1D steady-state local surface model are done
existence and uniqueness of the exact and numerical solutions, extra properties of the derivatives, and convergence of finite element schemes. Some numerical results of the two models treated separately are presented for a 2D plunging tape configuration.
TL;DR: This article conjugate time marching schemes with Finite Differences splittings into low and high modes in order to build fully explicit methods with enhanced temporal stability for the numerical solutions of PDEs.
Abstract: In this article, we conjugate time marching schemes with Finite Differences splittings into low and high modes in order to build fully explicit methods with enhanced temporal stability for the numerical solutions of PDEs. The main idea is to apply explicit schemes with less restrictive stability conditions to the linear term of the high modes equation, in order that the allowed time step for the temporal integration is only determined by the low modes. These conjugated schemes were developed in l10r for the spectral case and here we adapt them to the Finite Differences splittings provided by Incremental Unknowns, which steems from the Inertial Manifolds theory. We illustrate their improved capabilities with numerical solutions of Burgers equations, with uniform and nonuniform meshes, in dimensions one and two, when using modified Forward–Euler and Adams–Bashforth schemes. The resulting schemes use time steps of the same order of those used by semi-implicit schemes with comparable accuracy and reduced computational costs.
TL;DR: A new high spectral accuracy compact difference scheme is proposed here by constrained optimization of error in spectral space for discretizing first derivative for problems with non-periodic boundary condition, producing a scheme with the highest spectral accuracy among all known compact schemes.
Abstract: A new high spectral accuracy compact difference scheme is proposed here. This has been obtained by constrained optimization of error in spectral space for discretizing first derivative for problems with non-periodic boundary condition. This produces a scheme with the highest spectral accuracy among all known compact schemes, although this is formally only second-order accurate. Solution of Navier--Stokes equation for incompressible flows are reported here using this scheme to solve two fluid flow instability problems that are difficult to solve using explicit schemes. The first problem investigates the effect of wind-shear past bluff-body and the second problem involves predicting a vortex-induced instability.
TL;DR: In this article, adaptations of a first order method for the implementation at high order, which may employ either fixed or variable computation time steps, are presented, and comparisons are shown to establish comparisons between diverse numerical methods.
Abstract: The historical evolution of the equation of motion for a spherical particle in a fluid and the search for its general solution are recalled. The presence of an integral term that is nonzero under unsteady motion and viscous conditions allowed simple analytical or numerical solutions for the particle dynamics to be found only in a few particular cases. A general solution to the equation of motion seems to require the use of computational methods. Numerical schemes to handle the integral term of the equation of motion have already been developed. We present here adaptations of a first order method for the implementation at high order, which may employ either fixed or variable computation time steps. Some examples are shown to establish comparisons between diverse numerical methods.
TL;DR: A fast subtractional spectral algorithm for the solution of the Poisson equation and the Helmholtz equation which does not require an extension of the original domain and enjoys the following properties: fast convergence and high accuracy even when the computation employs a small number of collocation points.
Abstract: The paper presents a fast subtractional spectral algorithm for the solution of the Poisson equation and the Helmholtz equation which does not require an extension of the original domain. It takes O(N2 log N) operations, where N is the number of collocation points in each direction. The method is based on the eigenfunction expansion of the right hand side with integration and the successive solution of the corresponding homogeneous equation using Modified Fourier Method. Both the right hand side and the boundary conditions are not assumed to have any periodicity properties. This algorithm is used as a preconditioner for the iterative solution of elliptic equations with non-constant coefficients. The procedure enjoys the following properties: fast convergence and high accuracy even when the computation employs a small number of collocation points. We also apply the basic solver to the solution of the Poisson equation in complex geometries.
TL;DR: The present paper describes the use of compact upwind and compact central schemes in a Finite Volume formulation with an extension towards arbitrary meshes and a new formulation of artificial selective damping that is applicable on non-uniform Cartesian meshes.
Abstract: The present paper describes the use of compact upwind and compact central schemes in a Finite Volume formulation with an extension towards arbitrary meshes. The different schemes are analyzed and tested on several numerical experiments. A new formulation of artificial selective damping that is applicable on non-uniform Cartesian meshes is presented. Results are shown for a 1D advection equation, a 2D rotating Gaussian pulse and a subsonic inviscid vortical flow on uniform and non-uniform meshes and for a non-linear acoustic pulse.
TL;DR: It is shown that the moving mesh methods with the proposed monitor functions can effectively capture the free boundaries of the elliptic obstacle problems and reduce the numerical errors arising from thefree boundaries.
Abstract: The main objective of this work is to demonstrate that sharp a posteriori error estimators can be employed as appropriate monitor functions for moving mesh methods. We illustrate the main ideas by considering elliptic obstacle problems. Some important issues such as how to derive the sharp estimators and how to smooth the monitor functions are addressed. The numerical schemes are applied to a number of test problems in two dimensions. It is shown that the moving mesh methods with the proposed monitor functions can effectively capture the free boundaries of the elliptic obstacle problems and reduce the numerical errors arising from the free boundaries.
TL;DR: This paper is devoted to the approximation of a non standard Stokes problem by spectral methods: in addition to the pressure assigned on a part of a boundary, the tangential vorticity is given on another part of the boundary.
Abstract: This paper is devoted to the approximation of a non standard Stokes problem by spectral methods: in addition to the pressure assigned on a part of the boundary, the tangential vorticity is given on another part of the boundary. Several spectral discretizations are proposed and analysed. The inf-sup conditions, associated with the discretizations of this problem and with the spurious modes that follow from them, are thoroughly studied.