RK4 Scheme Research Articles

A fourth order Runge-Kutta type of exponential time differencing and triangular spectral element method for two dimensional nonlinear Maxwell's equations

In this paper, we study a numerical scheme to solve the nonlinear Maxwell's equations. The discrete scheme is based on the triangular spectral element method (TSEM) in space and the exponential time differencing fourth-order Runge-Kutta (ETDRK4) method in time. TSEM has the advantages of spectral accuracy and geometric flexibility. The ETD method involves exact integration of the linear part of the governing equation followed by an approximation of an integral involving the nonlinear terms. The RK4 scheme is introduced for the time integration of the nonlinear terms. The stability region of the ETDRK4 method is depicted. Moreover, the contour integral in the complex plan is utilized and improved to compute the matrix function required by the implementation of ETDRK4. The numerical results demonstrate that our proposed method is of exponential convergence with the order of basis function in space and fourth order accuracy in time.

Rapid variable-step computation of dynamic convolutions and Volterra-type integro-differential equations: RK45 Fehlberg, RK4

We introduce a novel, time-efficient adaptive Runge-Kutta computational scheme tailored for systematically solving linear and nonlinear Volterra-type Integro-Differential Equations (VTIDEs). This scheme is particularly effective for equations featuring a specific class of convolution memory, K(t,s)=aκ(t−s) within convolution integrals, where a,κ∈R. Such equations frequently arise in fields like viscoelasticity, population dynamics, epidemiology, control systems, financial mathematics, and neuroscience, where current states are influenced by historical data, especially in dynamical systems within physics. Our proposed scheme demonstrates remarkable computational efficiency, achieving a cost-effective computation time of O(Nt) for Nt iterations, a significant improvement over existing schemes for VTIDE, which typically exhibit complexities of O(Ntlog⁡Nt) and O(Nt(log⁡Nt))2. The key feature of our scheme lies in its ability to elegantly and efficiently handle convoluted integrals of varying complexity such as those including nonlinearities and derivatives. Specifically, our approach eliminates the need for backstage integration, allowing the main Runge-Kutta scheme to operate efficiently with only a few data points at each time step. This not only saves time but also prevents potential data storage issues while rendering numerical implementation straightforward and tractable. Moreover, this article introduces flexible discretizations for integration and differentiation, accommodating unequally spaced data points. We implement a fast RK45-Fehlberg algorithm (TE-RK45-Fehlberg), which we use to solve three examples of nonlinear VTIDE presented in this study. We further demonstrate the effectiveness of our approach by using an example, where we employ a novel parabolic step size generator function to efficiently generate step sizes for a computationally fast RK4 scheme (Flex-RK4). For all examples tackled using TE-RK45-Fehlberg and Flex-RK4, we generate plots that compare numerical data with exact solutions, providing insight into the accuracy and reliability of our approach. We also show how the scheme provides straightforward ways of transforming it into implicit forms and obtaining stability curves.

Comprehensive analysis of COVID-19 transmission dynamics: mathematical modeling, stability analysis, and optimal control strategies

This paper presents a mathematical model for comprehensively analyzing the transmission dynamics of COVID-19. We investigate the model’s various properties, such as positivity, boundedness, and the existence and uniqueness of solutions. Additionally, we calculate the basic reproductive number, denoted as R 0, to gauge the epidemic’s potential spread. Furthermore, we conduct a stability analysis to understand the long-term behavior of the model. Furthermore, we devised an optimal control strategy to effectively curb disease transmission. Using graphical analysis, we assess the impact of secondary infection rates and quarantine rates across different population groups. Finally, we compare our proposed numerical scheme with the well-known RK-4 scheme, emphasizing the NSFD scheme’s ability to maintain positivity, unlike the RK-4 scheme. Our numerical simulations offer strong evidence supporting the theoretical findings, demonstrating the effectiveness of our results.

Adaptation of nanofluids with magnetohydrodynamic Williamson fluid to enhance the thermal and solutal flow analysis with viscous dissipation: A numerical study

The Williamson nanofluid flow along with the thermal and solutal transport effects like enthalpy and viscous dissipation, has lots of applications in numerous engineering fields such as heat exchangers, automotive engineering, cooling systems, aerospace, and biomedical applications. The temperature-dependent viscosity has implications in the polymer processing, chemical processing, microfluidic devices, environmental fluid dynamics, and oil and gas industries. The current research deals with the numerical study of the magnetohydrodynamic flow of 2-dimensional Williamson nanofluid flowing on a stretched cylinder. The thermophysical property (viscosity) is assumed to depend on temperature for this flow system. The transmission analysis of solutal and heat processes, along with enthalpy and viscous dissipation, is also considered here. The boundary layer approach is used to address the main leading equations, and ODEs are generated from the main equations by assuming similarity transformations. The main novelty of this problem is the use of a multistep technique, which is the Adams-Bashforth Predictor and Corrector method, along with the RK-4 scheme and the secant method. The key parameters are used to illustrate the graphical and numerical behavior of the velocity, temperature, and concentration regions. The variations in the Williamson fluid coefficient cause the fluid velocity to increase, whereas alterations in the curvature parameter cause the velocity region to decrease. The temperature in a given area declines due to the impact of the Prandtl number. The fluids temperature shows augmented values in the occurrence of variation in the thermopherosis coefficient, Brownian motion coefficient, curvature parameter, and heat generation parameter. The curvature parameter, thermopherosis coefficient, and activation energy lead to an increase in the fluid concentration region, whereas a decrease in concentration is noted due to the Brownian motion coefficient, Damkholer number and Schmidt number.

Understanding the impact of magnetic dipole and variable viscosity on nanofluid flow characteristics over a stretching surface

The purpose of this research article is to investigate the impact of magnetic dipole on the flow of nano-fluids over the stretching surface with variable viscosity and thermal conductivity. Fluid fills the porous space. Heat transfer analysis is carried out in the presence of Newtonian heating. Water and ethylene glycol are used as base fluid while gamma alumina acts as a nanoparticle. System of ordinary differential equations is obtained by appropriate transformations. Numerical solutions are computed for the resulting nonlinear differential system by applying shooting method with RK-4 scheme. Impact of different parameters on the velocity and temperature profiles is examined. Computations for surface drag force and heat transfer rates are presented and examined for the influences of pertinent parameters. It is observed that fluid velocity reduces for larger viscosity parameter while opposite behavior is observed for temperature. Increasing values of ferro hydrodynamic interaction parameter and Curie temperature correspond to the enhancement in temperature. Surface drag force increases for larger values of nanoparticles volume fraction. Gamma alumina-ethylene glycol nano-fluid produce less surface drag force than gamma alumina-water nano-fluid. Heat transfer rate is increasing function of Prandtl number.

The stability analysis of a nonlinear mathematical model for typhoid fever disease

Typhoid fever is a contagious disease that is generally caused by bacteria known as Salmonella typhi. This disease spreads through manure contamination of food or water and infects unprotected people. In this work, our focus is to numerically examine the dynamical behavior of a typhoid fever nonlinear mathematical model. To achieve our objective, we utilize a conditionally stable Runge–Kutta scheme of order 4 (RK-4) and an unconditionally stable non-standard finite difference (NSFD) scheme to better understand the dynamical behavior of the continuous model. The primary advantage of using the NSFD scheme to solve differential equations is its capacity to discretize the continuous model while upholding crucial dynamical properties like the solutions convergence to equilibria and its positivity for all finite step sizes. Additionally, the NSFD scheme does not only address the deficiencies of the RK-4 scheme, but also provides results that are consistent with the continuous system's solutions. Our numerical results demonstrate that RK-4 scheme is dynamically reliable only for lower step size and, consequently cannot exactly retain the important features of the original continuous model. The NSFD scheme, on the other hand, is a strong and efficient method that presents an accurate portrayal of the original model. The purpose of developing the NSFD scheme for differential equations is to make sure that it is dynamically consistent, which means to discretize the continuous model while keeping significant dynamical properties including the convergence of equilibria and positivity of solutions for all step sizes. The numerical simulation also indicates that all the dynamical characteristics of the continuous model are conserved by discrete NSFD scheme. The theoretical and numerical results in the current work can be engaged as a useful tool for tracking the occurrence of typhoid fever disease.

Legendre wavelets based approach for the solution of type-2 fuzzy uncertain smoking model of fractional order

PurposeInvestigation of the smoking model is important as it has a direct effect on human health. This paper focuses on the numerical analysis of the fractional order giving up smoking model. Nonetheless, due to observational or experimental errors, or any other circumstance, it may contain some incomplete information. Fuzzy sets can be used to deal with uncertainty. Yet, there may be some inconsistency in the membership as well. As a result, the primary goal of this proposed work is to numerically solve the model in a type-2 fuzzy environment.Design/methodology/approachTriangular perfect quasi type-2 fuzzy numbers (TPQT2FNs) are used to deal with the uncertainty in the model. In this work, concepts of r2-cut at r1-plane are used to model the problem's uncertain parameter. The Legendre wavelet method (LWM) is then utilised to solve the giving up smoking model in a type-2 fuzzy environment.FindingsLWM has been effectively employed in conjunction with the r2-cut at r1-plane notion of type-2 fuzzy sets to solve the model. The LWM has the advantage of converting the non-linear fractional order model into a set of non-linear algebraic equations. LWM scheme solutions are found to be well agreed with RK4 scheme solutions. The existence and uniqueness of the model's solution have also been demonstrated.Originality/valueTo deal with the uncertainty, type-2 fuzzy numbers are used. The use of LWM in a type-2 fuzzy uncertain environment to achieve the model's required solutions is quite fascinating, and this is the key focus of this work.

Mathematical and numerical model for the malaria transmission: Euler method scheme for a malarial model

This research study has developed a mathematical model for malaria disease which is not only applicable for the case when the recovered humans return to the susceptible class, but also provides the directions for the case when the recovered humans also return to the infectious class. The model is simulated by using the Euler, Runge–Kutta-4 (RK-4), and nonstandard finite difference (NSFD) scheme. Firstly, the model is simulated by the Euler scheme and RK4 scheme and obtained graphical depiction for the endemic equilibrium as well as for the disease-free equilibrium (DFE). Then the mathematical model of malaria is simulated by an NSFD scheme and its graphical interpretation shows that it is suitable for all step sizes, i.e., it gives converging results even for very large step sizes. It is shown that the NSFD scheme is an unconditionally stable numerical scheme at a large step size. It is concluded that parameter [Formula: see text] is greater than unity in the disease manifestation of the landlord population in the long term and when the parameter [Formula: see text] is less than unity then the DFE is asymptotically stable.

A Higher-Order Galerkin Time Discretization and Numerical Comparisons for Two Models of HIV Infection.

Human immunodeficiency virus (HIV) infection affects the immune system, particularly white blood cells known as CD4+ T-cells. HIV destroys CD4+ T-cells and significantly reduces a human's resistance to viral infectious diseases as well as severe bacterial infections, which can lead to certain illnesses. The HIV framework is defined as a system of nonlinear first-order ordinary differential equations, and the innovative Galerkin technique is used to approximate the solutions of the model. To validate the findings, solve the model employing the Runge-Kutta (RK) technique of order four. The findings of the suggested techniques are compared with the results obtained from conventional schemes such as MuHPM, MVIM, and HPM that exist in the literature. Furthermore, the simulations are performed with different time step sizes, and the accuracy is measured at various time intervals. The numerical computations clearly demonstrate that the Galerkin scheme, in contrast to the Runge-Kutta scheme, provides incredibly precise solutions at relatively large time step sizes. A comparison of the solutions reveals that the obtained results through the Galerkin scheme are in fairly good agreement with the RK4 scheme in a given time interval as compared to other conventional schemes. Moreover, having performed various numerical tests for assessing the efficiency and computational cost (in terms of time) of the suggested schemes, it is observed that the Galerkin scheme is noticeably slower than the Runge-Kutta scheme. On the other hand, this work is also concerned with the path tracking and damped oscillatory behaviour of the model with a variable supply rate for the generation of new CD4+ T-cells (based on viral load concentration) and the HIV infection incidence rate. Additionally, we investigate the influence of various physical characteristics by varying their values and analysing them using graphs. The investigations indicate that the lateral system ensured more accurate predictions than the previous model.

Positivity preserving numerical method for epidemic model of hepatitis B disease dynamic with delay factor

This work attempts to study the numerical solution of nonlinear delayed Immunized Susceptible Latent Infected and Recovered (MSLIR) epidemic model of HBV disease. Reproduction number, equilibria and stability are discussed. Three different numerical techniques, Euler, RK-4 and the non-standard finite difference (NSFD) techniques are used for the numerical solution of the model. The proposed technique is independent of the size of the time step, while forward Euler and RK-4 depend on the size of a time step and retains all essential characteristics of the continuous MSLIR epidemic model like positivity and stability of equilibrium, while well-known forward Euler and RK-4 cannot sustain these characteristics. Therefore, the proposed (NSFD) technique becomes a more efficient and reliable numerical technique than the forward Euler and RK-4 scheme. Numerical simulations are presented for the validation of the obtained results.

Unsteady stagnation-point flow of CNTs suspended nanofluid on a shrinking/expanding sheet with partial slip: multiple solutions and stability analysis

The purpose of the study of single/multi-wall carbon nanotubes (SWCNTs/MWCNTs) mixed water-based nanofluid having unsteady stagnation-point flow on shrinking/expanding sheet with velocity and thermal slip effects is to decode the heat transfer mechanism to know the high cooling rate criteria. Governing boundary layer coupled partial differential equation (PDEs) are converted into ordinary ones. The transformed equations are numerically solved by shooting method with RK-4 scheme. The impacts of different parameters are described graphically and a comparison between current and previous results is made in tabular form. Existence of multiple solutions along with unique solution appears for specific cases of shrinking and expanding velocities. The investigation also reveals that SWCNT-nanoparticles have more dominating heat and momentum transfer rates than MWCNT-nanoparticles. Velocity slip delays the boundary layer separation, but maximum surface drag-force does not alter. Thermal slip and unsteadiness reduce heat transfer rate, whereas velocity slip enhances it. For high shrinking velocity compared to free stream velocity, the surface cooling rate drops down with Prandtl number and thermal boundary layer thickness significantly reduces for all types of nanofluids and for both solution branches. For both nanofluids, the temperature near the sheet decreases with thermal slip. For non-uniqueness of the similarity solution, a linear stability analysis is conducted and it verifies that upper branch of the obtained solutions is stable, while lower branch is unstable for high shrinking velocity and the unique solution is stable for expanding and low shrinking velocities.

A pseudo-spectral based efficient volume penalization scheme for Cahn–Hilliard equation in complex geometries

In this paper, we have developed an efficient volume penalization based diffuse-filtering scheme to solve variable mobility based Cahn–Hilliard equation in complex geometries. The main novelty of the work is that a dealiased pseudo-spectral scheme with immersed interface method (IIM) is proposed for solving any generalized concentration-dependent mobility function-based Cahn–Hilliard (CH) equation in complicated computational domains. An indicator function based mobility parameter is introduced to perform simulation of binary spinodal decomposition problem at a lower computational expense in complex geometries by solving a single phase field equation. Due to the smooth removal of high frequency Fourier components, the solution of the present RK4 based diffuse-filtering scheme does not display spurious currents when suitable low-pass filtering strategy and adequately resolved mobility indicator are incorporated. The traditional and memory optimized zero padding schemes are also implemented to show the comparative performance of different dealiasing schemes for the variable mobility based Cahn–Hilliard equation. It is found that the diffuse-filtering scheme displays reasonable accuracy similar to the zero padding based schemes but its average CPU time is significantly lower, which indicates better computational performance of the scheme for the variable mobility Cahn–Hilliard equation. Time variation of the characteristic length scale during spinodal decomposition of a binary mixture agrees well with the analytical prediction. The optimal three stage SSPRK3 temporal scheme is employed and it is found that time step size can be increased approximately 1.4 times than the classical RK4 scheme reducing total CPU time. Oscillation free numerical solution and conservation of order parameter are obtained for the complex geometry based spinodal decomposition problem. A radially-averaged structure factor is introduced to quantify resolution issues of the dealiasing schemes for the spinodal decomposition problem in different complex geometries.

Entropy generation optimization of unsteady radiative hybrid nanofluid flow over a slippery spinning disk

Entropy generation investigation of hybrid nanofluidic transport over an unsteady spinning disk is reported in this analysis. The magnetic influence, velocity slips, and thermal radiative effects are included within the flow. Ferrous oxide (Fe3O4) and graphene oxide (GO) are used as tiny nano ingredients, and water (H2O) is the base medium. The dimensional leading equations are settled to dimensionless nonlinear ordinary differential equations (ODEs) by significant similarity transformations. Then, classical RK-4 scheme with a shooting process has been initiated to execute the numerical simulation. The software MAPLE-18 is used to run the entire simulation with an indispensable accuracy rate. Several streamlines, graphs, and requisite tables are executed to divulge the parametric impact on the nanofluidic stream. Entropy generation–related figures are depicted for diverse parameters, and parametric effects on Bejan number are also analyzed. Moreover, the corresponding physical consignments like the measure of the frictional hindrance, heat transport are calculated and reviewed. The entropy generation augments for higher magnetic value but reduces for velocity slip, radiation, and nanoparticle concentration. Hybrid nanofluid gives a lower magnitude in entropy production as compared to the usual nanofluid. Magnetic parameter reduces the Bejan number, while slip factor and nanoparticle concentration amplify such effects. Heat transfer ultimately seems to increase for nanoparticle volume fraction, and the increase rate is 4.01685 for usual nanofluid, but it is 6.7557 for hybrid nanofluid. Also, the numerical fallouts address the possibility of using magnetized spinning disks in space engines and nuclear propulsion, and such a model conveys significant applications in heat transport improvement in so many industrial thermal management equipment and renewable energy systems.

Dynamical behaviour of HIV Infection with the influence of variable source term through Galerkin method

In this paper, we formulate the intricate process of Human Immunodeficiency Virus (HIV) infection in a mathematical model. In the formulation of the model, the density of CD4+ T-cells is categorized into healthy and infected classes while the density of free HIV viruses in the human host’s blood is considered a separate class. It is reported that when HIV enters a human’s body, it infects a large amount of CD4+ T-cells and unbalance the supply of new cells from the thymus. Therefore, we incorporate variable source term relying on the viral load for the supply of new cells instead of using a fixed value for the supply of these cells. We implement a continuous Galerkin Petrov time discretization scheme, particularly cGP(2)-scheme to visualize the dynamical behavior of the mentioned model and a detailed description of the effects of different physical parameters of interest are depicts graphically and discuss that how the level of healthy, infected CD4+ T-cells and free HIV particles varies concerning the emerging parameters in the model. In the proposed cGP(2)-method, two unknowns terms can be found by solving a 2×2 block system. This method is accurate of order 3 in the whole time interval and shows even superconvergence of order 4 in the discrete-time points. Furthermore, determine the solution of the model utilizing the fourth-order Runge-Kutta (RK4)-method and find out the absolute errors between the solutions obtained from both approaches. Finally, the validity and reliability of the proposed scheme are verified by comparing the numerical and graphical results with the RK-4 method. We examine the accuracy of the cGP(2)-sheme and observe that it produces more accurate results at relatively larger step sizes as in comparison to RK-4 scheme. Correspondence between the findings of cGP(2)-method and RK4-method reveal that the new technique is a promising tool for obtaining approximate solutions to the nonlinear systems of real-world problems.

Spherical shallow‐water wave simulation by a cubed‐sphere finite‐difference solver

AbstractWe consider the test suite for the shallow‐water (SW) equations on the sphere suggested by Paldor in earlier work. This series of tests consists of zonally propagating wave solutions on the full sphere. Two series of solutions are considered. The first series is referred to as “barotropic". It consists of an extension of the Rossby–Haurwitz test case. The second series, referred to as “baroclinic”, consists of a generalisation of the Matsuno solution to the linearized SW equations in an equatorial channel. The Hermitian Compact Cubed Sphere (HCCS) model which is used in this paper is a recently introduced SW solver on the sphere. The spatial approximation is a centred finite‐difference scheme based on high‐order differencing along great circles. The time stepping is performed by the explicit RK4 scheme or by an exponential scheme. For both barotropic and baroclinic test case series, the results show a very good agreement of the numerical solution with the analytic one, even for long time simulations.

Scalability of OpenFOAM Density-Based Solver with Runge–Kutta Temporal Discretization Scheme

Compressible density-based solvers are widely used in OpenFOAM, and the parallel scalability of these solvers is crucial for large-scale simulations. In this paper, we report our experiences with the scalability of OpenFOAM’s native rhoCentralFoam solver, and by making a small number of modifications to it, we show the degree to which the scalability of the solver can be improved. The main modification made is to replace the first-order accurate Euler scheme in rhoCentralFoam with a third-order accurate, four-stage Runge-Kutta or RK4 scheme for the time integration. The scaling test we used is the transonic flow over the ONERA M6 wing. This is a common validation test for compressible flows solvers in aerospace and other engineering applications. Numerical experiments show that our modified solver, referred to as rhoCentralRK4Foam, for the same spatial discretization, achieves as much as a 123.2% improvement in scalability over the rhoCentralFoam solver. As expected, the better time resolution of the Runge–Kutta scheme makes it more suitable for unsteady problems such as the Taylor–Green vortex decay where the new solver showed a 50% decrease in the overall time-to-solution compared to rhoCentralFoam to get to the final solution with the same numerical accuracy. Finally, the improved scalability can be traced to the improvement of the computation to communication ratio obtained by substituting the RK4 scheme in place of the Euler scheme. All numerical tests were conducted on a Cray XC40 parallel system, Theta, at Argonne National Laboratory.

RBF-FD Method for Some Dispersive Wave Equations and Their Eventual Periodicity

In this paper, we approximate the solution and also discuss the periodic behavior termed as eventual periodicity of solutions of (IBVPs) for some dispersive wave equations on a bounded domain corresponding to periodic forcing. The constructed numerical scheme is based on radial kernels and local in nature like finite difference method. The temporal variable is executed through RK4 scheme. Due to the local nature and sparse differentiation matrices our numerical scheme efficiently recovers the solution. The results achieved are validated and examined with other methods accessible in the literature.

Locally adaptive pseudo-time stepping for high-order Flux Reconstruction

This paper proposes a novel locally adaptive pseudo-time stepping convergence acceleration technique for dual time stepping which is a common integration method for solving unsteady low-Mach preconditioned/incompressible Navier-Stokes formulations. In contrast to standard local pseudo-time stepping techniques that are based on computing the local pseudo-time steps directly from estimates of the local Courant-Friedrichs-Lewy limit, the proposed technique controls the local pseudo-time steps using local truncation errors which are computed with embedded pair RK schemes. The approach has three advantages. First, it does not require an expression for the characteristic element size, which are difficult to obtain reliably for curved mixed-element meshes. Second, it allows a finer level of locality for high-order nodal discretisations, such as FR, since the local time-steps can vary between solution points and field variables. Third, it is well-suited to being combined with P-multigrid convergence acceleration. Results are presented for a laminar 2D cylinder test case at Re=100. A speed-up factor of 4.16 is achieved compared to global pseudo-time stepping with an RK4 scheme, while maintaining accuracy. When combined with P-multigrid convergence acceleration a speed-up factor of over 15 is achieved. Detailed analysis of the results reveals that pseudo-time steps adapt to element size/shape, solution state, and solution point location within each element. Finally, results are presented for a turbulent 3D SD7003 airfoil test case at Re=60,000. Speed-ups of similar magnitude are observed, and the flow physics is found to be in good agreement with previous studies.

The onset of nanofluid flow past a convectively heated shrinking sheet in presence of heat source/sink: A Lie group approach

The influence of uniform heat source or sink of Cu–water and Ag–water nanofluid flow over a convectively heated shrinking sheet has been studied. Lie group technique has been applied to translate highly non-linear governing partial differential equations into ordinary differential equation and then solved numerically by RK-4 scheme with shooting technique. The consequence of relevant parameters on the flow province has been discussed by means of graphical and tabular approach considering both assisting and opposing flow. Our analysis explores that Cu–water achieves high temperature even though Ag has high thermal conductivity with the impact of shrinking ratio parameter. Also temperature and heat transfer rate of Cu–water rise significantly for surface convection parameter.

Closure to “Evaluation of Explicit Numerical Solution Methods of the Muskingum Model” by Ali R. Vatankhah

Closure to “Evaluation of Explicit Numerical Solution Methods of the Muskingum Model” by Ali R. Vatankhah