Second-order Runge-Kutta Scheme Research Articles

Unified preserving properties of kinetic schemes.

The kinetic theory provides a physical basis for developing multiscale methods for gas flows covering a wide range of flow regimes. A particular challenge for kinetic schemes is whether they can capture the correct hydrodynamic behaviors of the system in the continuum regime (i.e., as the Knudsen number ε≪1) without enforcing kinetic scale resolution. At the current stage, the main approach to analyze such a property is the asymptotic preserving (AP) concept, which aims to show whether a kinetic scheme reduces to a solver for the hydrodynamic equationsas ε→0, such as the shock capturing scheme for the Euler equations. However, the detailed asymptotic properties of the kinetic scheme are indistinguishable when ε is small but finite under the AP framework. To distinguish different characteristics of kinetic schemes, in this paper we introduce the concept of unified preserving (UP) aiming at assessing asymptotic orders of a kinetic scheme by employing the modified equationapproach and Chapman-Enskon analysis. It is shown that the UP properties of a kinetic scheme generally depend on the spatial and temporal accuracy and closely on the interconnections among three scales (kinetic scale, numerical scale, and hydrodynamic scale) and their corresponding coupled dynamics. Specifically, the numerical resolution and specific discretization of particle transport and collision determine the flow physics of the scheme in different regimes, especially in the near continuum limit. As two examples, the UP methodology is applied to analyze the discrete unified gas-kinetic scheme and a second-order implicit-explicit Runge-Kutta scheme in their asymptotic behaviors in the continuum limit.

A coupled high-order implicit-explicit flux reconstruction lattice Boltzmann method for nearly incompressible thermal flows

In this paper, based on the double distribution function (DDF) model and the Boussinesq approximation, a high-order implicit-explicit flux reconstruction thermal lattice Boltzmann method (FRTLBM) is proposed for simulating nearly incompressible thermal flows. Two discrete velocity Boltzmann equations (DVBEs), one governing the flow field and the other governing the temperature field, are solved by using a high-order flux reconstruction scheme for spatial discretization and a second-order implicit-explicit Runge-Kutta scheme (IMEX RK) for time discretization in the generalized curvilinear coordinates with uniform and non-uniform grids. Numerical validations of the proposed method are implemented by simulating (a) porous plate problem, (b) natural convection in a square cavity, (c) Rayleigh-Bénard convection, (d) natural convection in a concentric annulus and (e) mixed heat transfer from a heated circular cylinder. Numerical results demonstrate that the present method can achieve the high-order accuracy and it is stable and accurate even at the relatively high Rayleigh number (Ra = 107 and 108). The present method is efficient due to the small amount of mesh, the simple algorithm and the time step unlimited by the relaxation time. Since the rescontruction is conducted in a single cell, the present method is compact for parallel computing. Moreover, the present method has a good adaptability of complicated geometries.

Simplified FDTD Formulations for Magnetostatic Biased Graphene Simulations

Simplified finite difference time domain (FDTD) formulations are presented for magnetostatic biased graphene simulations. In this respect, graphene's anisotropic conductivity is incorporated into the FDTD algorithm by using the second-order Runge-Kutta (RK) scheme with the minimal additional computational resources. Moreover, the complex frequency shifted perfectly matched layer is combined with the presented RK-FDTD formulations to model open-space graphene structures. Numerical tests are included to validate the given formulations.

Preconditioning methods for compressible flow CFD codes: Revisited

The present article aims to implement and investigate a different preconditioning method based in a three-dimensional in-house compressible CFD code that ensures the robustness and numerical stability to determine the flowfield considering low Mach number flow. The present preconditioning method involve two different methodologies developed by references [8, 32]. The CFD solver was developed to calculate the Euler and Navier-Stokes equations, numerically, for steady-state regime based on the cell-centered finite volume method (FVM) using Reynolds Averaged Navier-Stokes equations (RANS). The centered second-order scheme was used for the discretization of convective terms from momentum equations. The explicit second-order five-step Runge-Kutta scheme was employed for the time-marching procedure, using an implicit residual smoothing technique to enhance the numerical stability. A local preconditioning method was implemented due to its robustness in predicting low Mach number flows in a compressible CFD code environment. However, for low Mach number flows, near stagnation points, numerical perturbations were amplified generating a stiffness in the convergence rate, which provided an inaccurate solution and numerical stability degradation. Aiming to improve the preconditioning robustness, a flux function and a new limiter were applied to operate with the preconditioning technique based on a pressure sensor. Those corrections re-scale the eigenvectors and ensure the locality of the algorithm, which improves the numerical stability and guarantees the convergence for low-speed flows. The inviscid flow over a NACA 0012 airfoil for compressible and incompressible cases shown accurate and robust solutions. For a viscous flow over a flat plate case in the compressible and incompressible cases, the preconditioning technique purposed in this work supplied good numerical solution in agreement with the analytical solution.

A strongly S-stable low-dissipation and low-dispersion Runge-Kutta scheme for convection diffusion systems

Since the A-stability and the order of accuracy of time integration methods have been proven insufficient to guarantee the time-accurate simulations of convection diffusion systems, optimized implicit Runge-Kutta schemes with enhanced stability have been developed for stiff problems, and separately those with low-dispersion low-dissipation errors have been proposed for sensitive wave propagation phenomena. However, an implicit Runge-Kutta method is ideally preferred to address both disparate stiffness and various wave propagation characteristics that may be unknown in advance but often co-exist in complex systems, such as turbulent flows with multi-physical phenomena. Therefore, an optimized three-stage second-order diagonally implicit Runge-Kutta scheme with the strong S-stability and low-dispersion low-dissipation errors is derived in this study. Numerical benchmark tests show that overall this newly derived scheme has comprehensively the best performance among the well-known second-order implicit Runge-Kutta schemes. It reaches a second-order accuracy and produces accurate solutions for wave propagation phenomena, but is also stiffly first-order accurate and remains stable with very large time steps for strongly stiff problems.

High-Order Bicompact Schemes for Shock-Capturing Computations of Detonation Waves

An implicit scheme with splitting with respect to physical processes is proposed for a stiff system of two-dimensional Euler gas dynamics equations with chemical source terms. For the first time, convection is computed using a bicompact scheme that is fourth-order accurate in space and third-order accurate in time. This high-order bicompact scheme is L-stable in time. It employs a conservative limiting method and Cartesian meshes with solution-based adaptive mesh refinement. The chemical reactions are computed using an L-stable second-order Runge–Kutta scheme. The developed scheme is successfully tested as applied to several problems concerning detonation wave propagation in a two-species ideal gas with a single combustion reaction. The advantages of bicompact schemes over the popular MUSCL and WENO5 schemes as applied to shock-capturing computations of detonation waves are discussed.

Realizability-preserving DG-IMEX method for the two-moment model of fermion transport

Building on the framework of Zhang & Shu [1,2], we develop a realizability-preserving method to simulate the transport of particles (fermions) through a background material using a two-moment model that evolves the angular moments of a phase space distribution function f. The two-moment model is closed using algebraic moment closures; e.g., as proposed by Cernohorsky & Bludman [3] and Banach & Larecki [4]. Variations of this model have recently been used to simulate neutrino transport in nuclear astrophysics applications, including core-collapse supernovae and compact binary mergers. We employ the discontinuous Galerkin (DG) method for spatial discretization (in part to capture the asymptotic diffusion limit of the model) combined with implicit-explicit (IMEX) time integration to stably bypass short timescales induced by frequent interactions between particles and the background. Appropriate care is taken to ensure the method preserves strict algebraic bounds on the evolved moments (particle density and flux) as dictated by Pauli's exclusion principle, which demands a bounded distribution function (i.e., f∈[0,1]). This realizability-preserving scheme combines a suitable CFL condition, a realizability-enforcing limiter, a closure procedure based on Fermi-Dirac statistics, and an IMEX scheme whose stages can be written as a convex combination of forward Euler steps combined with a backward Euler step. The IMEX scheme is formally only first-order accurate, but works well in the diffusion limit, and — without interactions with the background — reduces to the optimal second-order strong stability-preserving explicit Runge-Kutta scheme of Shu & Osher [5]. Numerical results demonstrate the realizability-preserving properties of the scheme. We also demonstrate that the use of algebraic moment closures not based on Fermi-Dirac statistics can lead to unphysical moments in the context of fermion transport.

A Priori Error Analysis of a Discontinuous Galerkin Scheme for the Magnetic Induction Equation

Abstract We perform the error analysis of a stabilized discontinuous Galerkin scheme for the initial boundary value problem associated with the magnetic induction equations using standard discontinuous Lagrange basis functions. In order to obtain the quasi-optimal convergence incorporating second-order Runge–Kutta schemes for time discretization, we need a strengthened 4 / 3 {4/3} -CFL condition ( Δ ⁢ t ∼ h 4 / 3 {\Delta t\sim h^{4/3}} ). To overcome this unusual restriction on the CFL condition, we consider the explicit third-order Runge–Kutta scheme for time discretization. We demonstrate the error estimates in L 2 {L^{2}} -sense and obtain quasi-optimal convergence for smooth solution in space and time for piecewise polynomials with any degree l ≥ 1 {l\geq 1} under the standard CFL condition.

A fast immersed interface method for the Cahn–Hilliard equation with arbitrary boundary conditions in complex domains

A fast immersed boundary method for the Cahn–Hilliard equation is introduced. The decomposition of the fourth-order non-linear Cahn–Hilliard equation into a system of linear parabolic second-order equations allows to pose arbitrary Neumann or surface wetting conditions on the boundary. In space a finite-volume discretization on a regular Cartesian voxel grid allows the use of fast parabolic solvers via Fourier transform of arbitrary convergence order. For the time discretization, a second-order Runge–Kutta scheme is applied. The polynomial approximation of the chemical potential results in a numerical scheme that is unconditionally gradient-stable and allows large time steps. With an additional pre-conditioner for the linear system, the condition of linear system is minimized. By this the convergence is independent of both spatial discretization and time step size. This allows for the simulation of phase separation in large porous complex domains with three dimensions and over hundred million degrees of freedom while applying arbitrary boundary conditions and using large time steps.

An accurate predictor-corrector time integration method for structural dynamics

In this paper, a new accurate predictor-corrector time integration method is proposed, which has an appropriate stability domain. The dispersion error of the suggested approach is considerably smaller than that of the central difference scheme. Moreover, the presented technique introduces effectively numerical damping, which can be used to remove the spurious highfrequency components from the numerical responses. The authors’ approach has second-order accuracy for structures with and without damping. It is worth mentioning that the scheme solves the nonlinear problems without using iterative processes. The new algorithm is tested by using several linear and nonlinear numerical examples, and the outcomes are compared against the results of the central difference and explicit second-order Runge-Kutta schemes. The findings confirm the high accuracy and good performance of the suggested technique.

A SEMI-LAGRANGIAN TWO-LEVEL PRECONDITIONED NEWTON-KRYLOV SOLVER FOR CONSTRAINED DIFFEOMORPHIC IMAGE REGISTRATION.

We propose an efficient numerical algorithm for the solution of diffeomorphic image registration problems. We use a variational formulation constrained by a partial differential equation (PDE), where the constraints are a scalar transport equation. We use a pseudospectral discretization in space and second-order accurate semi-Lagrangian time stepping scheme for the transport equations. We solve for a stationary velocity field using a preconditioned, globalized, matrix-free Newton-Krylov scheme. We propose and test a two-level Hessian preconditioner. We consider two strategies for inverting the preconditioner on the coarse grid: a nested preconditioned conjugate gradient method (exact solve) and a nested Chebyshev iterative method (inexact solve) with a fixed number of iterations. We test the performance of our solver in different synthetic and real-world two-dimensional application scenarios. We study grid convergence and computational efficiency of our new scheme. We compare the performance of our solver against our initial implementation that uses the same spatial discretization but a standard, explicit, second-order Runge-Kutta scheme for the numerical time integration of the transport equations and a single-level preconditioner. Our improved scheme delivers significant speedups over our original implementation. As a highlight, we observe a 20× speedup for a two dimensional, real world multi-subject medical image registration problem.

A coupled two-dimensional numerical model for rapidly varying flow, sediment transport and bed morphology

ABSTRACTThis paper presents a coupled two-dimensional model that can produce a more stable numerical simulation of rapid bed evolution than the conventional decoupled model. To solve the coupled bed-load sediment transport terms using a Godunov-type central-upwind method, a novel scheme to estimate the bed-load fluxes which can produce more accurate results than the previously reported coupled model is proposed using a pair of local wave speeds different from those used for the flow. The two-dimensional shallow water equations are used to solve the flow velocities and water depth. The bed level is solved by an Exner-based equation containing bed-load sediment transport as numerical flux terms and sediment entrainment and deposition as source terms. Analytical formulas to compute the eigenvalues of the Jacobian matrix are developed. The linear reconstruction of variables with a multi-dimensional slope limiter, and the second-order Runge-Kutta scheme are employed to achieve higher accuracy in space and time. For the case of rapid bed-erosion, the accuracy and stability of the proposed coupled model are verified by several numerical tests.

Wet-dry moving boundary treatment for Runge-Kutta discontinuous Galerkin shallow water equation model

The wet-dry scheme for moving boundary treatment is implemented in the framework of discontinuous Galerkin shallow water equations. As a formulation of approximate Riemann solver, the HLL (Harten-Lax-van Leer) numerical fluxes are employed and the TVB (Total Variation Bounded) slope limiter is used for the removal of unnecessary oscillations. As benchmark test problems, the dam-break problems and the classical problem of periodic oscillation in the parabolic bowl are solved with linear triangular elements and second-order Runge-Kutta scheme. The results are compared with exact solutions and the numerical solutions of previous study. For a more practical application, the implicit Runge-Kutta scheme is employed for the bottom friction terms and the moving shoreline in a rectangular basin of varying slopes is simulated. In all case studies, good agreement is observed with exact solutions or other well-known numerical solutions.

Numerical Study on the Stability of Hypersonic Wake Flow over Blunt Body and Global Stability Analysis

The structural instabilities of base flow in hypersonic wake flow over sphere are investigated by numerical simulation in this paper. Numerical simulation method is second-order accuracy NND scheme in spatial direction and Second-order Runge-Kutta scheme in temporal direction. Wakes flow of sphere at M∞=2, Re=1.71E+6(based on the sphere radius) and at M∞=6, Re=1.71E+6 are simulated. Stable solutions are obtained and there are a primary separation and a secondary separation zone in flow over sphere at M∞=2 and M∞=6. Later, without any arbitrary disturbance imposed, the structural instability still occurs after a certain critical time. The evolving process of the structure of flow over sphere is a periodic behavior in the forepart, but it evolves into a non-periodic behavior after a critical time at M∞=2. In the period time, the topological structures of the flow field over sphere and their evolving process at M∞=6 are similar to that of M∞=2, but there is non-periodic behavior observed at M∞=6 in spite of long calculation time. In the case of sphere, the results by global stability theory, numerical results and results of power spectrum are compared each other. The global stability analysis method can give a good prediction result. This study result indicates that the present numerical simulation method and global stability analysis method can be used for the study of the stability of base flow and confirms that hypersonic flow at the base is also instability.

Numerical solution of the FENE-CR model in complex flows

A finite difference technique for solving the FENE-CR (Finite Extendable Non-linear Elastic – Chilcott and Rallison) closure constitutive model in complex flows has been developed and tested. The governing equations are solved using a Marker-and-Cell type method on a staggered grid. The momentum equation is integrated employing an implicit method while the FENE-CR constitutive equation is approximated by a second-order Runge–Kutta scheme. To demonstrate that the numerical technique can cope with complex flows governed by the FENE-CR model, three flow problems were analysed: the fully-developed channel flow, the 2D cross-slot flow and the impacting drop problem. The analytic solution for fully-developed channel flow of FENE-CR fluids with a solvent viscosity is also presented for validation purposes. This flow problem is used to verify the numerical method and to quantify its accuracy by comparing numerical results of fully-developed channel flow with the analytic solution. The second flow is employed to assess whether the numerical methodology is capable of capturing the purely-elastic instabilities predicted in the literature for 2D cross-slot confined flows. Additionally, the complex free surface flow corresponding to the filling of a 2D cross geometry has also been investigated. The last problem concerns the flow dynamics of a FENE-CR fluid drop impacting on a rigid surface, which allows the assessment of the capability of the model to deal with free surfaces. The effects of varying the Reynolds number, the Weissenberg number and the finite extensibility of the polymer molecules (L2) on the resulting flow patterns are analysed.

Time stepping in discontinuous Galerkin method

The time discretization in the Discontinuous Galerkin (DG) scheme has been traditionally based on the Total Variation Diminishing (TVD) second-order Runge-Kutta (RK2) scheme. Computational efficiency and accuracy with the Euler Forward (EF) and the TVD second-order RK2 time stepping schemes in the DG method are investigated in this work. Numerical tests are conducted with the scalar Burgers equation, 1-D and 2-D shallow water flow equations. The maximum Courant number or time step size required for stability for the EF scheme and RK2 scheme with different slope limiters are compared. Numerical results show that the slope limiters affect the stability requirement in the DG method. The RK2 scheme is generally more diffusive than the EF scheme, and the RK2 scheme allows larger time step sizes. The EF scheme is found to be more efficient and accurate than the RK2 scheme in the DG method in computation.

Three-dimensional transient complex free surface flows: Numerical simulation of XPP fluid

In this paper we present a finite difference MAC-type approach for solving three-dimensional viscoelastic incompressible free surface flows governed by the eXtended Pom–Pom (XPP) model, considering a wide range of parameters. The numerical formulation presented in this work is an extension to three-dimensions of our implicit technique [Journal of Non-Newtonian Fluid Mechanics 166 (2011) 165–179] for solving two-dimensional viscoelastic free surface flows. To enhance the stability of the numerical method, we employ a combination of the projection method with an implicit technique for treating the pressure on the free surfaces. The differential constitutive equation of the fluid is solved using a second-order Runge–Kutta scheme. The numerical technique is validated by performing a mesh refinement study on a pipe flow, and the numerical results presented include the simulation of two complex viscoelastic free surface flows: extrudate-swell problem and jet buckling phenomenon.

A finite volume method for scalar conservation laws with stochastic time–space dependent flux functions

We propose a new finite volume method for scalar conservation laws with stochastic time–space dependent flux functions. The stochastic effects appear in the flux function and can be interpreted as a random manner to localize the discontinuity in the time–space dependent flux function. The location of the interface between the fluxes can be obtained by solving a system of stochastic differential equations for the velocity fluctuation and displacement variable. In this paper we develop a modified Rusanov method for the reconstruction of numerical fluxes in the finite volume discretization. To solve the system of stochastic differential equations for the interface we apply a second-order Runge–Kutta scheme. Numerical results are presented for stochastic problems in traffic flow and two-phase flow applications. It is found that the proposed finite volume method offers a robust and accurate approach for solving scalar conservation laws with stochastic time–space dependent flux functions.

Simulating finger phenomena in porous media with a moving finite element method

The non-equilibrium Richards equation is solved using a moving finite element method in this paper. The governing equation is discretized spatially with a standard finite element method, and temporally with second-order Runge–Kutta schemes. A strategy of the mesh movement is based on the work by Li et al. [R.Li, T.Tang, P.W. Zhang, A moving mesh finite element algorithm for singular problems in two and three space dimensions, Journal of Computational Physics, 177 (2002) 365–393]. A Beckett and Mackenzie type monitor function is adopted. To obtain high quality meshes around the wetting front, a smoothing method which is based on the diffusive mechanism is used. With the moving mesh technique, high mesh quality and high numerical accuracy are obtained successfully. The numerical convergence and the advantage of the algorithm are demonstrated by a series of numerical experiments.

差分格子ボルツマン法によるトンネル圧縮波の発生と伝播計算(流体工学,流体機械)

Direct numerical simulation of compression wave produced when a high-speed train enters a tunnel, distortion of the compression wave front as it travels in the tunnel, radiation of micropressure wave from the tunnel exit are performed using the finite difference lattice Boltzmann method. The discrete Boltzmann equation for the 3D39Q thermal BGK model is solved in three-dimensional space using a second-order Runge-Kutta scheme in time and a third-order-upwind finite difference scheme in space. The arbitrary Lagrangian-Eulerian formulation (ALE) is applied to model the interaction of the moving train nose and the tunnel portal. Detailed numerical calculations were carried out for axisymmetric trains with the blockage of 0.2 and various nose profiles entering a long circular cylindrical tunnel with straight and stepwise flared portals. The predicted compression wave profiles are found to be in good agreement with linear theory predictions obtained from the analytic expression derived by Howe. It is shown that the distortion of the compression wave front is consistent with the time-domain computation of one-dimensional Burgers equation. Longer nose profiles and tunnel entrance with flared portals are confirmed not only to decrease the initial steepness of the compression wave front but also to counteract the effect of nonlinear steepening.