Second-order Runge-Kutta Research Articles

Nonlinear Hermitian generalized hygrothermoelastic stress and wave propagation analyses of thick FGM spheres exhibiting temperature, moisture, and strain-rate material dependencies

The present article is dedicated to the dynamic stress and displacement distributions and hygrothermoelastic wave propagation and reflection responses of FG hollow spheres subjected to thermomechanical shocks. The strain-rate and temperature dependencies of the material properties are accounted for. Hence, the material properties of the sphere are dependent on coordinates, time, loading rate, temperature, and ambient humidity. Furthermore, material degradation due to moisture absorption is considered. It is the first time that such a complex and more realistic combination is taken into account in the wave propagation analysis. The nonlinear coupled Lord-Shulman-type generalized hygrothermoelasticity equations that are developed by the inclusion of the moisture absorption state variable into the free energy function, are solved by using a Galerkin-type finite element method, iterative solution algorithm, and a second-order Runge-Kutta time integration procedure. Results are extracted by using the C1-continuous Hermitian rather than common C0-continuous Lagrangian elements to guarantee exact continuity of the stresses at the mutual boundaries of the elements and preclude the numerical locking phenomenon. Comprehensive sensitivity analyses including the effects of various factors are performed. Results reveal the significant effects of the temperature, strain-rate, and moisture absorption on both the material properties and constitutive law and consequently, on the transient stress distribution and the hygrothermoelastic wave propagation/reflection phenomenon. Furthermore, the results confirm that in the FG structures, the thermal and stress wavefronts travel with variable and time-dependent speeds.

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.

Application of Transformation Matrices to the Solution of Population Balance Equations

The development of algorithms and methods for modelling flowsheets in the field of granular materials has a number of challenges. The difficulties are mainly related to the inhomogeneity of solid materials, requiring a description of granular materials using distributed parameters. To overcome some of these problems, an approach with transformation matrices can be used. This allows one to quantitatively describe the material transitions between different classes in a multidimensional distributed set of parameters, making it possible to properly handle dependent distributions. This contribution proposes a new method for formulating transformation matrices using population balance equations (PBE) for agglomeration and milling processes. The finite volume method for spatial discretization and the second-order Runge–Kutta method were used to obtain the complete discretized form of the PBE and to calculate the transformation matrices. The proposed method was implemented in the flowsheet modelling framework Dyssol to demonstrate and prove its applicability. Hence, it was revealed that this new approach allows the modelling of complex processes involving materials described by several interconnected distributed parameters, correctly taking into consideration their interdependency.

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.

Explicit integration methods for constitutive equations of a mean-stress dependent elastoviscoplastic model: impact on structural finite element analyses

The strong dependent behavior of semi-crystalline polymers can lead to the use of simplified material laws in finite element structural calculations for reasons of robustness to the detriment of the quantitative response of the models. This work focuses on numerical integration methods as a solution to overcome the possible convergence and robustness limitations of mean-stress dependent elastoviscoplastic material laws, typical of the semi-crystalline polymers’ mechanical behavior. What is proposed here is a rational application of three explicit integration methods (fourth- and second-order Runge–Kutta method, a hybrid schema between Runge–Kutta, and Euler method) in engineering structural calculations, which provide a reliable solution for constitutive models of semi-crystalline polymer. These methods are examined for structure creep test and tensile test, in comparison with experimental data. The investigations have been done in terms of the stability toward convergence, the accuracy of results, the plastic consistency, and CPU time efficiency. This work, proposes an easy implementation of integration methods in any computational finite element code. It also provides a flexible modular implementation which is applicable to any different constitutive equations. An integration step sub-division technique is recommended. It is a powerful technique to improve the convergence of solution and accuracy of result by damping oscillation around stress Gauss point integration solution. The results obtained illustrate the effect of numerical integration schemas on structural analysis and provide an insight into select suitable method.

Numerical solutions of the coupled unsteady nonlinear convection-diffusion equations based on generalized finite difference method

In this paper, the meshless Generalized Finite Difference Method (GFDM) in conjunction with the second-order explicit Runge-Kutta method (RK2 method) is presented to solve coupled unsteady nonlinear convection-diffusion equations (CDEs). Compared with the conventional Euler method, the RK2 method not only has higher accuracy but also reduces the possibility of numerical oscillation in time discretization, especially for the nonlinear and coupled cases. The generalized finite difference method, which is a localized collocation method, is famous for its simplicity and adaptability in the numerical solution of partial differential equations. Benefiting from Taylor series and moving least squares, its partial derivatives can be formed by a series of surrounding space points. In comparison with traditional finite difference methods, the proposed GFDM is free of mesh and available for irregular discretization nodes. In this study, the stencil selection algorithms are introduced to choose the stencil support of a certain node from the whole discretization nodes. Error analysis and numerical investigations are presented to demonstrate the effectiveness of the proposed GFDM for solving the coupled linear and nonlinear unsteady convection-diffusion equations. Then it is successfully applied to three benchmark examples of the coupled unsteady nonlinear CDEs encountered in the double-diffusive natural convection process, chemotaxis-haptotaxis model of cancer invasion, and thermo-hygro coupling model of concrete.

On large start-up error of BDF2

The second-order backward-difference (BDF2) method requires solutions at two previous time steps and is often started with the first-order backward-difference (BDF1) method. The initial time step is typically taken to be small in order to minimize the effect of the first-order error. However, contrary to expectations, the solution computed in this way generates a very large error for stiff problems and the error grows as the initial time step is further reduced. As shown in this note, the problem is originated from the fact that the variable-coefficient BDF2 method combined with any first- or higher-order one-step method is asymptotically equivalent to the trapezoidal method and therefore not L-stable. Based on the study presented, it is strongly recommended that BDF2 method be started with a self-starting second-order implicit Runge-Kutta method, which guarantees second-order accuracy and L-stability at no additional cost.

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 precise direct parametric interpolation method for NURBS-toolpath generation

Non-uniform rational B-spline (NURBS) interpolator has many advantages over conventional linear/circular interpolator. However, as there is a nonlinear corresponding relation between the curve parameter and the arc displacement of the NURBS curve, it is hard to accurately calculate the parameter of the next interpolation point, thus resulting in a deviation between the desired and the actual feed rate, i.e., feed rate fluctuation. A large feed rate fluctuation may cause machine-tool vibration, which is bad for the cutter life and the machining quality. Therefore, aiming at reducing the feed rate fluctuation, this paper presents a high-precision parametric interpolation method, which features a second-order Runge-Kutta algorithm with parameter compensation. First, a second-order Runge-Kutta method is employed to compute the initial interpolation-point parameter. After that, the parameter is compensated using first-order Taylor's expansion of the feed rate-fluctuation function, thus enhancing the calculation precision of the next interpolation point parameter. Experimental results demonstrate the advantage of the presented method.

A high accurate hamiltonian nodal position finite element method for spatial cable structures undergoing long-term large overall motion

This paper addresses the challenges faced by the error accumulation over long-term numerical calculation for dynamic modeling of spatial flexible cable structures undergoing large translational and rotational motion. A high accurate Hamiltonian nodal position finite element method is proposed to deal with the challenges. The new nodal position finite element discrete formulation is derived by Hamiltonian theory and Green strain theory with full expression of global stiffness matrices without additional simplifications. Symplectic difference algorithm is built for numerical solution to optimally preserve the energy, momenta and area (volume) of the phase space. Forth-order closed Newton-Cotes numerical integration is applied to calculate the aerodynamic drag force. The Symplectic conservation feature of the proposed method is validated by the dynamics of a long-period classical pendulum. The numerical accuracy and stability of the proposed method are validated by LS-DYNA simulations for a flexible polyethylene rubber conical pendulum, the experiments of a three-dimensional circularly towed cable and the experiments of a free swing cable. The present algorithm is compared with the conventional second-order Runge-Kutta algorithm. All validations and comparisons indicate that the proposed method is stable and highly accurate.

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.

An evaluation model for the effect of adsorption boundary on pollutant degradation in water areas

Abstract A two-dimensional water quality numerical model has been developed to analyse the influence of adsorption boundary on the variation of water pollutant degradation. The characteristic finite element method is adopted to solve the discrete pollutant concentration equations, so the time-monotonicity of concentration is ensured in the convection-dominated flow field with various adsorption conditions. The second-order Runge–Kutta method is adopted to track the upstream concentration source points along the streamline, so the precision of the calculated pollutant concentration is improved even under the streamline sharp-bending condition. The flow roughness and pollutant degradation coefficient are formulated by elements according to the actual situation, so the model can quantitatively analyse the effect of various ecological restoration schemes on purification capacity in water areas. The adaptability and robustness of the model were verified by a typical engineering case, where the calculated results coincide with the measured data.

A discontinuous Galerkin approach for conservative modeling of fully nonlinear and weakly dispersive wave transformations

This work extends a robust second-order Runge–Kutta Discontinuous Galerkin (RKDG2) method to solve the fully nonlinear and weakly dispersive flows, within a scope to simultaneously address accuracy, conservativeness, cost-efficiency and practical needs. The mathematical model governing such flows is based on a variant form of the Green–Naghdi (GN) equations decomposed as a hyperbolic shallow water system with an elliptic source term. Practical features of relevance (i.e. conservative modeling over irregular terrain with wetting and drying and local slope limiting) have been restored from an RKDG2 solver to the Nonlinear Shallow Water (NSW) equations, alongside new considerations to integrate elliptic source terms (i.e. via a fourth-order local discretization of the topography) and to enable local capturing of breaking waves (i.e. via adding a detector for switching off the dispersive terms). Numerical results are presented, demonstrating the overall capability of the proposed approach in achieving realistic prediction of nearshore wave processes involving both nonlinearity and dispersion effects within a single model.

Numerical solutions of waves-current interactions by generalized finite difference method

In this paper, a meshless numerical wave flume, based on the generalized finite difference method (GFDM), is adopted to accurately and efficiently simulate the interactions of water waves and current. The GFDM, a newly-developed meshless method, is truly free from mesh generation and numerical quadrature. The proposed meshless numerical wave flume is the combination of the GFDM, the second-order Runge–Kutta method, the semi-Lagrangian approach, the sponge layer and the ramping function. The problems of wave-current interactions in flumes with horizontal and inclined bottoms are accurately and stably investigated by the proposed meshless scheme, respectively. The changes of waveform can be obviously found, while the cases of coplanar, opposing and no currents are stably simulated. Besides, the distribution of steady current in the flume with inclined bottom, which is governed by an inverse Cauchy problem, is acquired by the GFDM in a stable manner. Numerical results of wave-current interactions are compared with other solutions to verify the accuracy of the proposed meshless scheme. Additionally, different parameters of the proposed meshless numerical scheme are examined to validate the consistency and stability of the proposed numerical wave flume for solutions of wave-current interactions.

Chaotic predation scheme for age-clustered one predator–one prey Lotka–Volterra

Lotka–Volterra differential equations deal with modeling of predator and prey populations and interrelation of population sizes in a continuous domain. Since all populations, be they predators or prays, are only the estimates in consequence of sampling process, this paper initially concerns with discretization schemes of one predator–prey Lotka–Volterra systems. Second-order Runge–Kutta approximation is used to discretize the original differential equations for flexibility to manipulate the system parameters. Subsequent to discretization, a novel predation scheme is introduced to enhance the rationalization of original model with age cluster-based and detailed rules which are based on reproductivity, predation ability, predation essentiality, food provision and consumption. Aging is also simulated with transitive structure of the algorithms that the alive individuals are getting older and changing clusters after the predation scheme is operated. Experiments revealed that our model exhibits not only fluctuations like the original model but also stable trajectories and fractal structure depending on the model parameters. Therefore, the main novelty of this paper briefly is the discovery of chaotic one predator–one prey system exhibiting chaotic behavior for a narrow interval and revealing a strange attractor which is very unique.

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.

Global stability analysis of lifted diffusion flames

This work describes the development of a method for the global hydrodynamic stability analysis of diffusion flames. The low-Machnumber (LMN) Navier–Stokes (NS) equations for reacting flows are solved together with a transport equation for the mixture fraction describing the local composition of the fluid. The equations are solved by the spectral-element code NEK5000 with Legendre polynomial reconstruction of degree twelve and second-order accurate Runge-Kutta time integration scheme. In order to compute the base flow for the stability analysis, a selective frequency damping approach has been employed. The global stability analysis has been performed by a matrix-free time-stepper algorithm applied to the LMN-NS equations, using an Arnoldi method to compute the most unstable modes. Moreover, a numerical linearization of the governing equation is employed, which allows one to study the stability of diffusion flames without the direct evaluation and storage of the linearized operator. Therefore, a remarkable reduction of the storage capacity is achieved and a more flexible numerical approach is obtained. The numerical model has been validated by comparison with the results for the axisymmetric diffusion flame available in the literature.

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.

Advancing parabolic operators in thermodynamic MHD models: Explicit super time-stepping versus implicit schemes with Krylov solvers

We explore the performance and advantages/disadvantages of using unconditionally stable explicit super time-stepping (STS) algorithms versus implicit schemes with Krylov solvers for integrating parabolic operators in thermodynamic MHD models of the solar corona. Specifically, we compare the second-order Runge-Kutta Legendre (RKL2) STS method with the implicit backward Euler scheme computed using the preconditioned conjugate gradient (PCG) solver with both a point-Jacobi and a non-overlapping domain decomposition ILU0 preconditioner. The algorithms are used to integrate anisotropic Spitzer thermal conduction and artificial kinematic viscosity at time-steps much larger than classic explicit stability criteria allow. A key component of the comparison is the use of an established MHD model (MAS) to compute a real-world simulation on a large HPC cluster. Special attention is placed on the parallel scaling of the algorithms. It is shown that, for a specific problem and model, the RKL2 method is comparable or surpasses the implicit method with PCG solvers in performance and scaling, but suffers from some accuracy limitations. These limitations, and the applicability of RKL methods are briefly discussed.

Generalized finite difference method for two-dimensional shallow water equations

A novel meshless numerical scheme, based on the generalized finite difference method (GFDM), is proposed to accurately analyze the two–dimensional shallow water equations (SWEs). The SWEs are a hyperbolic system of first-order nonlinear partial differential equations and can be used to describe various problems in hydraulic and ocean engineering, so it is of great importance to develop an efficient and accurate numerical model to analyze the SWEs. According to split-coefficient matrix methods, the SWEs can be transformed to a characteristic form, which can easily present information of characteristic in the correct directions. The GFDM and the second-order Runge-Kutta method are adopted for spatial and temporal discretization of the characteristic form of the SWEs, respectively. The GFDM is one of the newly-developed domain-type meshless methods, so the time-consuming tasks of mesh generation and numerical quadrature can be truly avoided. To use the moving-least squares method of the GFDM, the spatial derivatives at every node can be expressed as linear combinations of nearby function values with different weighting coefficients. In order to properly cooperate with the split-coefficient matrix methods and the characteristic of the SWEs, a new way to determine the shape of star in the GFDM is proposed in this paper to capture the wave transmission. Numerical results and comparisons from several examples are provided to verify the merits of the proposed meshless scheme. Besides, the numerical results are compared with other solutions to validate the accuracy and the consistency of the proposed meshless numerical scheme.