Group Preserving Scheme Research Articles

Forward and inverse river contaminant transport modeling using group preserving scheme

Environmental concerns have necessitated the development of computational models predicting pollutant dispersion in natural water systems. Due to the ill-posed nature of the inverse contaminant transport equation, solving this equation using all stable and convergent inverse methods is impossible. Factors such as river geometry, unsteady and non-uniform flow, and tidal influences add to the complexity of the inverse problem. These factors have prompted the evaluation of Group Preserving Scheme for environmental applications. The inverse solution method derives a general equation for solving ordinary differential equations by addressing a dynamical system at negative time steps, ensuring convergence. Three test cases have been presented to evaluate Backward Group Preserving Scheme (BGPS). These have included validation using observational data from Missouri River, inverse simulation in a tidal river, and sensitivity analysis of parameters such as pollutant patterns, advection, dispersion, and decay coefficients. The dataset includes calibrated data from Missouri River, which demonstrated high accuracy, with Mean Relative Error (MRE) ranging from 2.8% to 5.0% for the inverse model. Under tidal conditions, accuracy decreases over time but remains robust, with a Nash–Sutcliffe efficiency of 0.77–0.96 and an MRE of 0.9%–5.9%. Sensitivity analysis revealed optimal model performance for Péclet numbers greater than 500. The model performed best with gradual, wide-peaked pollutant patterns and moderate decay rates (Damköhler number between 5 and 10). BGPS proves effective for transport simulation and concentration history reconstruction in complex rivers, including those with tidal influences, offering a robust tool for water contamination analysis in various flow conditions.

Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators

In the paper, we first develop a novel automatically energy-preserving scheme (AEPS) for the undamped and unforced single and multi-coupled Duffing equations by recasting them to the Lie-type systems of ordinary differential equations. The AEPS can automatically preserve the energy to be a constant value in a long-term free vibration behavior. The analytical solution of a special Duffing–van der Pol equation is compared with that computed by the novel group-preserving scheme (GPS) which has fourth-order accuracy. The main novelty is that we constructed the quadratic forms of the energy equations, the Lie-algebras and Lie-groups for the multi-coupled Duffing oscillator system. Then, we extend the GPS to the damped and forced Duffing equations. The corresponding algorithms are developed, which are effective to depict the long term nonlinear vibration behaviors of the multi-coupled Duffing oscillators with an accuracy of O(h4) for a small time stepsize h.

New wave solutions, exact and numerical approximations to the nonlinear Klein–Gordon equation

This study investigates the nonlinear Klein–Gordon equation (KGE). We successfully construct some new topological kink-type, non-topological, singular solitons, periodic waves and singular periodic wave solutions to this nonlinear model by using the extended ShGEEM, rational sine-cosine extended (ERSC), and sinh-cosh (ERSCh) methods. In addition, a numerical method for solving the KGE is described in this paper. We use a combination of two numerical techniques called fictitious time integration method and the group preserving scheme (GPS). Fictitious time integration method converts the main equation into a new problem then the GPS is used to gain the numerical solutions. Few experiments are provided to successfully demonstrate the correctness of the approach.

A Novel Coordinated Control Strategy for Parallel Hybrid Electric Vehicles during Clutch Slipping Process

For parallel hybrid electric vehicles (HEVs), the clutch serves as a vital enabling actuator element during mode transitions. The expected drivability and smoothness of parallel HEVs are difficult to be achieve owing to the neglect of clutch-torque-induced disturbance and different response characteristics of power sources during clutch slipping. To address this issue, this paper proposes a novel control strategy to coordinate the engine and motor during the clutch slipping process. A sliding mode control strategy based on a group-preserving scheme was applied to control the motor. The vehicle dynamic equation was constructed by the sliding surface with the Lagrange function. The equation solutions obtained by introducing the Runge–Kutta method were used as motor control inputs. Meanwhile, an adaptive PI controller was designed to regulate engine speed for the reduction in the speed difference of the clutch. The hardware-in-the-loop simulations were conducted to validate the outstanding performance of the proposal strategy. The verification results indicate that the proposed strategy not only reduces the vehicle jerk and frictional losses effectively, but also improves vehicle driving comfort and reliability.

Numerical approximations and conservation laws for the Sine-Gordon equation

In the present work, an efficient Lie group integrator called group preserving scheme (GPS) is utilized for the numerical treatment of the Sine-Gordon problems. With the help of a suitable transformation, we turn the main problem into a new one and consequently solve the converted problem using GPS. Indeed, preserving the Lie group structure under discretization has a significant impress on regaining a proper behavior in error minimization. So, by sharing the geometric structure and invariance of the original ODEs, new approaches can be derived, which are more strong, precise, and stable than the conventional numerical methods. Some examples are provided to show the consistency of the approach. Moreover, we construct the conserved vectors (CV) for the governing equation by means of a new theorem of conservation.

A simple, efficient and accurate new Lie--group shooting method for solving nonlinear boundary value problems

The present paper provides a new method for numerical solution of nonlinear boundary value problems. This method is a combination of group preserving scheme (GPS) and a shooting--like technique which takes advantage of two powerful methods for solving nonlinear boundary value problems. This method is very effective to search unknown initial conditions. To demonstrate the computational efficiency, the mentioned method is implemented for some nonlinear exactly solvable differential equations including strongly nonlinear Bratu equation, nonlinear reaction--diffusion equation and one singular nonlinear boundary value problem. It is also applied successfully on two nonlinear three--point boundary value problems and a third--order nonlinear boundary value problem which the exact solutions of this problems are unknown. The examples show the power of method to search for unique solution or multiple solutions of nonlinear boundary value problems with high computational speed and high accuracy. In the test problem 5 a new branch of solutions is found which shows the power of the method to search for multiple solutions and indicates that the method is successful in cases where purely analytic methods are not.

A Lie group integrator to solve the hydromagnetic stagnation point flow of a second grade fluid over a stretching sheet

<abstract><p>In the present paper, a Lie-group integrator, based on $ GL(4, \mathbb{R}) $ has been newly constructed to consider the flow characteristics in an electrically conducting second grade fluid over a stretching sheet. Present method which have a very fast convergence, permits us to explore some missing initial values at the left-end. Accurate initial values can be achieved when the determined target equation is valid, and then we can apply the group preserving scheme (GPS) as a geometric approach to obtain a rather accurate numerical solution. Finally, effects of magnetic parameter, viscoelastic parameter, stagnation point flow and stretching of the sheet parameters are illustrated.</p></abstract>

The Equal-Norm Multiple-Scale Trefftz Method for Solving the Nonlinear Sloshing Problem with Baffles

In this paper, the equal-norm multiple-scale Trefftz method combined with the implicit Lie-group scheme is applied to solve the two-dimensional nonlinear sloshing problem with baffles. When considering solving sloshing problems with baffles by using boundary integral methods, degenerate geometry and problems of numerical instability are inevitable. To avoid numerical instability, the multiple-scale characteristic lengths are introduced into T-complete basis functions to efficiently govern the high-order oscillation disturbance. Again, the numerical noise propagation at each time step is eliminated by the vector regularization method and the group-preserving scheme. A weighting factor of the group-preserving scheme is introduced into a linear system and then used in the initial and boundary value problems (IBVPs) at each time step. More importantly, the parameters of the algorithm, namely, the T-complete function, dissipation factor, and time step, can obtain a linear relationship. The boundary noise interference and energy conservation are successfully overcome, and the accuracy of the boundary value problem is also improved. Finally, benchmark cases are used to verify the correctness of the numerical algorithm. The numerical results show that this algorithm is efficient and stable for nonlinear two-dimensional sloshing problems with baffles.

New solutions of fractional-order Burger-Huxley equation

A simple and powerful numerical method is proposed for solving the time-fractional Burger-Huxley equation (TFBH) equation within Caputo type fractional derivative. A fictitious coordinate θ is applied into the equation in order to convert the dependent variable u(x,t) into a new variable with an extra dimension. In the new space with the added fictitious dimension, a combination of the method of line and group preserving scheme (GPS) is introduced to find the approximate solutions. The reliability and accuracy of this scheme has been shown through some examples of the TFBH equation.

Numerical comparisons based on new NCP-functions

We consider studying numerical comparisons based on variants of new NCP-functions denoted by . The nonlinear programming (NLP) can be converted to nonlinear complementarity problem (NCP) by employing the Karush-Kuhn Tucker (KKT) optimality conditions. One of the most popular ways to solve NCP is Lagrangian globalization (LG) method by transforming NCP as a system of nonsmooth (semismooth) equations. The second one is a novel method named the fictitious time integration method (FTIM). We reformulate NCP as a system of nonlinear algebraic equations (NAEs) and then construct an ordinary differential equation (ODE) by utilizing time-like functions. A group preserving scheme (GPS) is a set of ODEs which is a tool for systematically reformulating into the nonlinear dynamical system (NDS). Afterward, the NDS can be manipulated to numerical equations through activating the Lorentz group SO 0(n, 1) and its Lie algebra SO 0(n, 1). The FTIM will be operated on this numerical equation in numerical simulations for getting approximation solutions. All of the numerical experiments are carried out in performance profile theories. The comparisons of new NCP-functions by utilizing FTIM and LG method will be discussed in numerical experiments. Lastly, an accurate test of both FTIM and LG method is going to be committed by performance profile concepts as well.

An ordinary differential equation approach for nonlinear programming and nonlinear complementary problem

We consider an ordinary differential equation (ODE) approach for solving non- linear programming (NLP) and nonlinear complementary problem (NCP). The Karush- Kuhn Tucker (KKT) optimality conditions can be converted to NCP. Based on the Fischer-Burmeister (FB) function and the Natural-Residual (NR) function are obtained the new NCP-functions. A special technique is employed to reformulate of the NCP as the system of nonlinear algebraic equations (NAEs) later on reformulated once more by force. of an original time-like function into an ODE. Afterwards, a group preserving scheme (GPS) is a package to reformulate an ODE into the new numerical equation in a way the ODEs system is designed into a nonlinear dynamical system (NDS) and is continued to a discovery the new numerical equation through activating the Lorentz group SO0(n, 1) and its Lie algebra so(n, 1). Lastly, the fictitious time integration method (FTIM) is utilized into this new numerical equation to determine an approximation solution at the numerical experiments area.

On Numerical Solution Of The Time Fractional Advection-Diffusion Equation Involving Atangana-Baleanu-Caputo Derivative

Abstract A powerful algorithm is proposed to get the solutions of the time fractional Advection-Diffusion equation(TFADE): A B C D 0 + , t β u ( x , t ) = ζ u x x ( x , t ) − κ u x ( x , t ) + $^{ABC}\mathcal{D}_{0^+,t}^{\beta}u(x,t) =\zeta u_{xx}(x,t)- \kappa u_x(x,t)+$ F(x, t), 0 < β ≤ 1. The time-fractional derivative A B C D 0 + , t β u ( x , t ) $^{ABC}\mathcal{D}_{0^+,t}^{\beta}u(x,t)$ is described in the Atangana-Baleanu Caputo concept. The basis of our approach is transforming the original equation into a new equation by imposing a transformation involving a fictitious coordinate. Then, a geometric scheme namely the group preserving scheme (GPS) is implemented to solve the new equation by taking an initial guess. Moreover, in order to present the power of the presented approach some examples are solved, successfully.

On the MHD boundary layer flow with diffusion and chemical reaction over a porous flat plate with suction/blowing: two reliable methods

In this paper, a Lie-group integrator based on $$GL_4(\mathbb {R})$$ and the reproducing kernel functions has been constructed to investigate the flow characteristics in an electrically conducting second-grade fluid over a stretching sheet. Accurate initial values can be achieved when the target equation is matched precisely, and then, we can apply the group preserving scheme (GPS) to get a rather accurate results. On the other hand, the reproducing kernel method (RKM) is successfully applied to the underlying equation with convergence analysis. We show exact and approximate solutions by series in the reproducing kernel space. We use a bounded linear operator in the reproducing kernel space to get the solutions by the reproducing kernel method. Comparison of these two methods demonstrates the power and reliability. Finally, effects of magnetic parameter, viscoelastic parameter, stagnation-point flow, and stretching of the sheet parameters are illustrated.

On numerical solution of the time-fractional diffusion-wave equation with the fictitious time integration method

In this work we offer a robust numerical algorithm based on the Lie group to solve the time-fractional diffusion-wave (TFDW) equation. Firstly, we use a fictitious time variable \( \xi\) to convert the related variable u(x, t) into a new space with one extra dimension. Then by using a composition of the group preserving scheme (GPS) and a semi-discretization of new variable, we approximate the solutions of the problem. Finally, various numerical experiments are performed to illustrate the power and accuracy of the given method.

Numerical solution to the telegraph equation via the geometric moving Kriging meshfree method

Finding an appropriate shape function plays an important role in meshfree methods. Shape functions extracted from the moving Kriging (MK) interpolation is of recent interest for many researchers. We applied this technique to the weak form of the telegraph equation (TE) in space coordinates. Then we get a system of second-order ordinary differential equations w.r.t. the time variable. Some new variables are introduced to convert this system into a system of first-order ordinary differential equations. Then the resultant system is solved by the group preserving scheme (GPS). Examples are provided to demonstrate the power and accuracy of the proposed method for the one-dimensional hyperbolic TE.

Numerical treatment on one-dimensional hyperbolic telegraph equation by the method of line-group preserving scheme

In the present paper, we introduce a powerful geometric numerical method for the non-homogenous one-dimensional telegraph equation (TE). A combination of semi-discretization and geometric integrator, group preserving scheme (GPS), is offered to obtain the numerical solutions. The stability analysis and convergence of the utilized method are investigated successfully. Power and accuracy of proposed method have been confirmed by some numerical experiments done for the TE.

Solving the Lane–Emden Equation within a Reproducing Kernel Method and Group Preserving Scheme

We apply the reproducing kernel method and group preserving scheme for investigating the Lane–Emden equation. The reproducing kernel method is implemented by the useful reproducing kernel functions and the numerical approximations are given. These approximations demonstrate the preciseness of the investigated techniques.

The combination of meshless method based on radial basis functions with a geometric numerical integration method for solving partial differential equations: Application to the heat equation

In this paper a new scheme is investigated for solving partial differential equations via combination of radial basis functions (RBFs) and group preserving scheme (GPS), which takes advantage of two powerful methods. In this method, we use Kansas approach to approximate the spatial derivatives and then we apply GPS method to approximate first-order time derivative. An advantage of the developed method is that it can be applied to problems with non-regular geometrical domains. To show the efficiency of this method, some heat equations are solved in one, two and three dimension spaces. The two-dimensional version of heat equation on different geometries such as the rectangular, triangular and circular domains is solved. The three-dimensional case is solved on the cubical and spherical domains. To show the high accuracy of the method, a comparison study of the present method and method used in the paper of Dehghan [1] is given.

The Lie-group method based on radial basis functions for solving nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation in arbitrary domains

The aim of this paper is to introduce a new numerical method for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation. This method is combination of group preserving scheme (GPS) with radial basis functions (RBFs), which takes advantage of two powerful methods, one as geometric numerical integration method and the other meshless method. Thus, we introduce this method as the Lie-group method based on radial basis functions (LG–RBFs). In this method, we use Kansas approach to approximate the spatial derivatives and then we apply GPS method to approximate first-order time derivative. One of the important advantages of the developed method is that it can be applied to problems on arbitrary geometry with high dimensions. To demonstrate this point, we solve nonlinear GBBMB equation on various geometric domains in one, two and three dimension spaces. The results of numerical experiments are compared with analytical solutions and the method presented in Dehghan et al. (2014) to confirm the accuracy and efficiency of the presented method.

Numerical Solution of a Nonlinear Fractional Integro-Differential Equation by a Geometric Approach

Numerical solution of a Riemann–Liouville fractional integro-differential boundary value problem with a fractional nonlocal integral boundary condition is studied based on a numerical approach which preserve the geometric structure on the Lorentz Lie group. A fictitious time $$\tau $$ is used to transform the dependent variable y(t) into a new one $$u(t,\tau ):=(1+\tau )^{\gamma }y(t)$$ in an augmented space, where $$0<\gamma \le 1$$ is a parameter, such that under a semi-discretization method and use of a Newton-Cotes quadrature rule the original equation is converted to a system of ODEs in the space $$(t,\tau )$$ and the obtained system is solved by the Group Preserving Scheme (GPS). Some illustrative examples are given to demonstrate the accuracy and implementation of the method.