Classical Fourth-order Runge Kutta Scheme Research Articles

A cost-effective semi-implicit method for the time integration of fully compressible reacting flows with stiff chemistry

We present a simple method to remove the stiffness associated with the chemical source terms in the fully compressible Navier-Stokes equations when the classical fourth order Runge-Kutta scheme is used.

Positivity and global stability preserving NSFD schemes for a mixing propagation model of computer viruses

In this work, nonstandard finite difference (NSFD) schemes preserving the essential qualitative properties including positivity and stability of a mixing propagation model of computer viruses are proposed and analyzed for the first time. Especially, the model under consideration possesses the equilibrium points which are not only locally asymptotically stable but also globally asymptotically stable. Because of this reason we propose a new approach to prove theoretically that the global asymptotic stability of the original model is preserved by the proposed NSFD schemes. This approach is based on the Lyapunov stability theorem and its extension in combination with a theorem on the global stability of discrete-time nonlinear cascade systems. The important result is that we obtain NSFD schemes which are dynamically consistent with the continuous model. Some numerical examples are performed to support the theoretical results. The results show that there is a good agreement between the numerical simulations and the established theoretical results. Furthermore, the numerical simulations also indicate that the proposed NSFD schemes are suitable and effective to solve the continuous model, whereas, the standard numerical schemes such as the Euler scheme, the classical fourth order Runge–Kutta scheme are not dynamically consistent with the continuous model and consequently, they fail to reflect the correct behavior of the continuous model.

On the global asymptotic stability of a hepatitis B epidemic model and its solutions by nonstandard numerical schemes

Very recently, a hepatitis B epidemic model with saturated incidence rate has been proposed and analyzed in Khan et al. (Chaos Solitons Fractals 124:1–9, 2019). The local asymptotic stability of the disease endemic equilibrium (DEE) point of model has been established theoretically but its global asymptotic stability has not been studied. In this paper, we present a mathematically rigorous analysis for the global asymptotic stability of the DEE point of the model. More precisely, we prove that if the DEE point exists, then it is not only locally asymptotically stable but also globally asymptotically stable. Furthermore, we present an alternative proof for the global stability of the disease-free equilibrium point. The main result is that we obtain the complete global stability of the hepatitis B virus model. Besides, we construct nonstandard finite difference (NSFD) schemes preserving the essential qualitative properties of the continuous model. These properties include the positivity of solutions and the stability of the model. Dynamical properties of the proposed schemes are rigorously investigated by mathematical analyses and numerical simulations. Finally, numerical simulations are performed to confirm the validity of the theoretical results as well as the advantages of the NSFD schemes. Numerical simulations also indicate that the NSFD schemes are appropriate and effective to solve the continuous model. Meanwhile, the standard finite difference schemes such as the Euler scheme, the classical fourth-order Runge–Kutta scheme cannot preserve the essential properties of the continuous model. Consequently, they can generate numerical approximations which are completely different from the solutions of the model.

A fourth-order accurate finite difference method to evaluate the true transient behaviour of natural convection flow in enclosures

PurposeModelling accurately the transient behaviour of natural convection flow in enclosures been a challenging task because of a variety of numerical errors which have limited achieving the higher order temporal accuracy. A fourth-order accurate finite difference method in both space and time is proposed to overcome these numerical errors and accurately model the transient behaviour of natural convection flow in enclosures using vorticity–streamfunction formulation.Design/methodology/approachFourth-order wide stencil formula with appropriate one-sided difference extrapolation technique near the boundary is used for spatial discretisation, and classical fourth-order Runge–Kutta scheme is applied for transient term discretisation. The proposed method is applied on two transient case studies, i.e. convection–diffusion of a Gaussian Pulse and Taylor Vortex flow having analytical solution.FindingsError magnitude comparison and rate of convergence analysis of the proposed method with these analytical solutions establish fourth-order accuracy and prove the ability of the proposed method to truly capture the transient behaviour of incompressible flow. Also, to test the transient natural convection flow behaviour, the algorithm is tested on differentially heated square cavity at high Rayleigh number in the range of 103-108, followed by studying the transient periodic behaviour in a differentially heated vertical cavity of aspect ratio 8:1. An excellent comparison is obtained with standard benchmark results.Research limitations/implicationsThe developed method is applied on 2D enclosures; however, the present methodology can be extended to 3D enclosures using velocity–vorticity formulations which shall be explored in future.Originality/valueThe proposed methodology to achieve fourth-order accurate transient simulation of natural convection flows is novel, to the best of the authors’ knowledge. Stable fourth-order vorticity boundary conditions are derived for boundary and external boundary regions. The selected case studies for comparison demonstrate not only the fourth-order accuracy but also the considerable reduction in error magnitude by increasing the temporal accuracy. Also, this study provides novel benchmark results at five different locations within the differentially heated vertical cavity of aspect ratio 8:1 for future comparison studies.

Nonstandard finite difference schemes for a general predator–prey system

In this paper, we transform a continuous-time predator–prey system with general functional response and recruitment for both species into a discrete-time model by nonstandard finite difference (NSFD) scheme. We prove theoretically and confirm by numerical simulations that the constructed NSFD schemes preserve the essential qualitative properties including positivity and stability of the continuous model for any finite step size. We also show that the standard finite difference schemes such as the Euler scheme, the second order Runge-Kutta scheme and the classical fourth order Runge-Kutta scheme cannot preserve the properties of the continuous model for large step sizes. They can generate the numerical solutions which are completely different from the solutions of the continuous model. Especially, the global stability of a non-hyperbolic equilibrium point of the constructed NSFD schemes is proved by the Lyapunov stability theorem. The performed numerical simulations confirm the validity of the obtained theoretical results as well as the advantages of NSFD schemes over standard ones.

A modified phase change pseudopotential lattice Boltzmann model

Amongst all the multiphase models in the lattice Boltzmann (LB) community, the pseudopotential model has been the most popular approach due to its simplicity and high-efficiency. Recently a number of liquid-vapour phase change models were also proposed based on the pseudopotential LB model. Our study finds that most of the published pseudopotential phase change models rely on an entropy-based energy equation, while the entropy-based energy equation is derived with the equation of state of ideal gas. That means this entropy-based energy equation is not completely suitable for multiphase flow which applies non-ideal equation of state for the phase separation simulation. Therefore a new phase change LB model is proposed in this work, where an improved pseudopotential multiphase model [32] and a modified energy equation which is solved in the classical fourth-order Runge-Kutta scheme are coupled in a hybrid scheme. The results show that the numerical simulation can capture the basic liquid-vapour phase change features. The D2 law for droplet evaporation is validated and the square of diameter variation is in good agreement with experimental data. Moreover, the three boiling stages (nucleate boiling, transition boiling and film boiling) are accomplished using the modified model, and the corresponding transient heat fluxes are presented.

Insight into the boundary layer flow of non-Newtonian Eyring-Powell fluid due to catalytic surface reaction on an upper horizontal surface of a paraboloid of revolution

Bonnet of a car, the upper surface of a pointed bullet and upper surface of the pointed part of an aircraft are typical examples of an upper horizontal surface of a paraboloid of revolution (uhspr). However, the flow of some fluids past these kinds of objects fit the description of Eyring-Powell fluid flow. Theoretical investigation of two-dimensional Eyring-Powell fluid flow over such object which is neither cone/wedge nor horizontal/vertical is investigated. It is assumed that the flow of Eyring-Powell fluid is induced by catalytic surface reaction and stretching fluid layers at the free stream. The numerical solutions of the governing equation are obtained using classical fourth order Runge-Kutta scheme together with shooting techniques. The impacts of the most important parameters on the flow are presented. It is concluded that the maximum velocity of the flow is ascertained when the flow is characterized as Newtonian fluid flow. On the surface of uhspr, rapid increase and suppress in the temperature distribution and concentration with an increase in the magnitude of one of the Eyring-Powell fluid parameters are guaranteed. A significant fall in the local skin friction coefficients is ascertained due to rise in the magnitude of thickness parameter.

An explicit hybrid method for multi-term fractional differential equations based on Adams and Runge-Kutta schemes

An explicit hybrid numerical algorithm is proposed by combining Adams-type and Runge-Kutta (RK) methods to solve fractional differential equations (FDEs). A general kind of multi-term FDEs is transformed into combined systems containing both FDEs and ordinary differential equations (ODEs). An explicit Adams-type scheme is constructed to integrate the FDEs. With the obtained solutions of FDEs, the ODEs are explicitly solved by the classical fourth-order RK scheme. Several types of FDEs are presented to validate the proposed method. Compared with an extended predictor corrector method, the presented method can provide more accurate results by an order of magnitude at the expense of one half of computational resources. With such high precision and efficiency, the presented method is applied in numerical simulations of long-term dynamic responses such as asymptotic periodic or limit cycle solutions of fractional dynamic systems.

Simulation of forced deformable bodies interacting with two-dimensional incompressible flows: Application to fish-like swimming

We present an efficient algorithm for simulation of deformable bodies interacting with two-dimensional incompressible fluid flows. The temporal and spatial discretizations of the Navier–Stokes equations in vorticity stream-function formulation are based on classical fourth-order Runge–Kutta scheme and compact finite differences, respectively. Using a uniform Cartesian grid we benefit from the advantage of a new fourth-order direct solver for the Poisson equation to ensure the incompressibility constraint down to machine zero over an optimal grid. For introducing a deformable body in fluid flow, the volume penalization method is used. A Lagrangian structured grid with prescribed motion covers the deformable body which is interacting with the surrounding fluid due to the hydrodynamic forces and the torque calculated on the Eulerian reference grid. An efficient law for controlling the curvature of an anguilliform fish, swimming toward a prescribed goal, is proposed which is based on the geometrically exact theory of nonlinear beams and quaternions. Validation of the developed method shows the efficiency and expected accuracy of the algorithm for fish-like swimming and also for a variety of fluid/solid interaction problems.

Fourier Spectral Methods for Solving Some Nonlinear Partial Differential Equations

The spectral collocation or pseudospectral (PS) methods (Fourier transform methods) combined with temporal discretization techniques to numerically compute solutions of some partial differential equations (PDEs). In this paper, we solve the Korteweg-de Vries (KdV) equation using a Fourier spectral collocation method to discretize the space variable, leap frog and classical fourth-order Runge-Kutta scheme (RK4) for time dependence. Also, Boussinesq equation is solving by a Fourier spectral collocation method to discretize the space variable, finite difference and classical fourth-order RungeKutta scheme (RK4) for time dependence. Our implementation employs the Fast Fourier Transform (FFT) algorithm.

Comparative study of travelling-wave and numerical solutions for the coupled short pulse (CSP) equation

The Lie symmetry analysis is performed for the coupled short plus (CSP) equation. We derive the infinitesimals that admit the classical symmetry group. Five types arise depending on the nature of the Lie symmetry generator. In all types, we find reductions in terms of system of ordinary differential equations, and exact solutions of the CSP equation are derived, which are compared with numerical solutions using the classical fourth-order Runge—Kutta scheme.

Parallel High-Order Integrators

In this work we discuss a class of defect correction methods which is easily adapted to create parallel time integrators for multicore architectures and is ideally suited for developing methods which can be order adaptive in time. The method is based on integral deferred correction (IDC), which was itself motivated by spectral deferred correction by Dutt, Greengard, and Rokhlin [BIT, 40 (2000), pp. 241-266]. The method presented here is a revised formulation of explicit IDC, dubbed revisionist IDC (RIDC), which can achieve $p$th-order accuracy in “wall-clock time” comparable to a single forward Euler simulation on problems where the time to evaluate the right-hand side of a system of differential equations is greater than latency costs of interprocessor communication, such as in the case of the $N$-body problem. The key idea is to rewrite the defect correction framework so that, after initial start-up costs, each correction loop can be lagged behind the previous correction loop in a manner that facilitates running the predictor and $M=p-1$ correctors in parallel on an interval which has $K$ steps, where $K\gg p$. We prove that given an $r$th-order Runge-Kutta method in both the prediction and $M$ correction loops of RIDC, then the method is order $r\times(M+1)$. The parallelization in RIDC uses a small number of cores (the number of processors used is limited by the order one wants to achieve). Using a four-core CPU, it is natural to think about fourth-order RIDC built with forward Euler, or eighth-order RIDC constructed with second-order Runge-Kutta. Numerical tests on an $N$-body simulation show that RIDC methods can be significantly faster than popular Runge-Kutta methods such as the classical fourth-order Runge-Kutta scheme. In a PDE setting, one can imagine coupling RIDC time integrators with parallel spatial evaluators, thereby increasing the parallelization. The ideas behind RIDC extend to implicit and semi-implicit IDC and have high potential in this area.

Very High-Order Spatially Implicit Schemes for Computational Acoustics on Curvilinear Meshes

A high-order compact-differencing and filtering algorithm, coupled with the classical fourth-order Runge–Kutta scheme, is developed and implemented to simulate aeroacoustic phenomena on curvilinear geometries. Several issues pertinent to the use of such schemes are addressed. The impact of mesh stretching in the generation of high-frequency spurious modes is examined and the need for a discriminating higher-order filter procedure is established and resolved. The incorporation of these filtering techniques also permits a robust treatment of outflow radiation condition by taking advantage of energy transfer to high-frequencies caused by rapid mesh stretching. For conditions on the scatterer, higher-order one-sided filter treatments are shown to be superior in terms of accuracy and stability compared to standard explicit variations. Computations demonstrate that these algorithmic components are also crucial to the success of interface treatments created in multi-domain and domain-decomposition strategies. Fo...

VERY HIGH-ORDER SPATIALLY IMPLICIT SCHEMES FOR COMPUTATIONAL ACOUSTICS ON CURVILINEAR MESHES

A high-order compact-differencing and filtering algorithm, coupled with the classical fourth-order Runge–Kutta scheme, is developed and implemented to simulate aeroacoustic phenomena on curvilinear geometries. Several issues pertinent to the use of such schemes are addressed. The impact of mesh stretching in the generation of high-frequency spurious modes is examined and the need for a discriminating higher-order filter procedure is established and resolved. The incorporation of these filtering techniques also permits a robust treatment of outflow radiation condition by taking advantage of energy transfer to high-frequencies caused by rapid mesh stretching. For conditions on the scatterer, higher-order one-sided filter treatments are shown to be superior in terms of accuracy and stability compared to standard explicit variations. Computations demonstrate that these algorithmic components are also crucial to the success of interface treatments created in multi-domain and domain-decomposition strategies. For three-dimensional computations, special metric relations are employed to assure the fidelity of the scheme in highly curvilinear meshes. A variety of problems, including several benchmark computations, demonstrate the success of the overall computational strategy.