In this paper we discuss, an initial-boundary value problem of hyperbolic type with first derivative with respect to x. The asymptotic solution is constructed and its uniform validity is proved under weader compatibil...In this paper we discuss, an initial-boundary value problem of hyperbolic type with first derivative with respect to x. The asymptotic solution is constructed and its uniform validity is proved under weader compatibility conditions. Then we develop an exponentially fitted difference scheme and establish discrete energy inequality. Finally, we prove that the solution of difference problem uniformly converges to the solution of the original problem.展开更多
In this article,we improve the order of precision of the two-dimensional Poisson equation by combining extrapolation techniques with high order schemes.The high order solutions obtained traditionally generate non-spar...In this article,we improve the order of precision of the two-dimensional Poisson equation by combining extrapolation techniques with high order schemes.The high order solutions obtained traditionally generate non-sparse matrices and the calculation time is very high.We can obtain sparse matrices by applying compact schemes.In this article,we compare compact and exponential finite difference schemes of fourth order.The numerical solutions are calculated in quadruple precision(Real*16 or extended precision)in FORTRAN language,and iteratively obtained until reaching the round-off error magnitude around 1.0E−32.This procedure is performed to ensure that there is no iteration error.The Repeated Richardson Extrapolation(RRE)method combines numerical solutions in different grids,determining higher orders of accuracy.The main contribution of this work is based on a process that initializes with fourth order solutions combining with RRE in order to find solutions of sixth,eighth,and tenth order of precision.The multigrid Full Approximation Scheme(FAS)is also applied to accelerate the convergence and obtain the numerical solutions on the fine grids.展开更多
This study develops a high-order computational scheme for analyzing unsteady tangent hyperbolic fluid flow with variable thermal conductivity,thermal radiation,and coupled heat andmass transfer effects.Amodified twost...This study develops a high-order computational scheme for analyzing unsteady tangent hyperbolic fluid flow with variable thermal conductivity,thermal radiation,and coupled heat andmass transfer effects.Amodified twostage Exponential Time Integrator is introduced for temporal discretization,providing second-order accuracy in time.A compact finite difference method is employed for spatial discretization,yielding sixth-order accuracy at most grid points.The proposed framework ensures numerical stability and convergence when solving stiff,nonlinear parabolic systems arising in fluid flow and heat transfer problems.The novelty of the work lies in combining exponential integrator schemes with compact high-order spatial discretization,enabling accurate and efficient simulations of tangent hyperbolic fluids under complex boundary conditions,such as oscillatory plates and varying thermal conductivity.This approach addresses limitations of classical Euler,Runge–Kutta,and spectral methods by significantly reducing numerical errors(up to 45%)and computational cost.Comprehensive parametric studies demonstrate how viscous dissipation,chemical reactions,the Weissenberg number,and the Hartmann number influence flow behaviour,heat transfer,and mass transfer.Notably,heat transfer rates increase by 18.6%with stronger viscous dissipation,while mass transfer rates rise by 21.3%with more intense chemical reactions.The real-world relevance of the study is underscored by its direct applications in polymer processing,heat exchanger design,radiative thermal management in aerospace,and biofluid transport in biomedical systems.The proposed scheme thus provides a robust numerical framework that not only advances the mathematical modelling of non-Newtonian fluid flows but also offers practical insights for engineering systems involving tangent hyperbolic fluids.展开更多
Singularly perturbed boundary value problem with nonlocal conditions is examined. The appopriate solution exhibits boundary layer behavior for small positive values of the perturbative parameter. An exponentially fitt...Singularly perturbed boundary value problem with nonlocal conditions is examined. The appopriate solution exhibits boundary layer behavior for small positive values of the perturbative parameter. An exponentially fitted finite difference scheme on a non-equidistant mesh is constructed for solving this problem. The uniform convergence analysis in small parameter is given. Numerical example is provided, too.展开更多
This paper applies exponentially fitted trapezoidal scheme to a stochastic oscillator. The scheme is convergent with mean-square order 1 and symplectic. Its numerical solution oscillates and the second moment increase...This paper applies exponentially fitted trapezoidal scheme to a stochastic oscillator. The scheme is convergent with mean-square order 1 and symplectic. Its numerical solution oscillates and the second moment increases linearly with time. The numerical example verifies the analysis of the scheme.展开更多
An efficient and accurate exponential wave integrator Fourier pseudospectral （EWI-FP） method is proposed and analyzed for solving the symmetric regularized-long-wave （SRLW） equation, which is used for modeling the...An efficient and accurate exponential wave integrator Fourier pseudospectral （EWI-FP） method is proposed and analyzed for solving the symmetric regularized-long-wave （SRLW） equation, which is used for modeling the weakly nonlinear ion acoustic and space-charge waves. The numerical method here is based on a Gautschi-type exponential wave integrator for temporal approximation and the Fourier pseudospectral method for spatial discretization. The scheme is fully explicit and efficient due to the fast Fourier transform. Numerical analysis of the proposed EWI-FP method is carried out and rigorous error estimates are established without CFL-type condition by means of the mathematical induction. The error bound shows that EWI-FP has second order accuracy in time and spectral accuracy in space. Numerical results are reported to confirm the theoretical studies and indicate that the error bound here is optimal.展开更多
This paper analyzes a population model with time-dependent advection and an autocatalytic-type growth.As opposed to a logistic growth where the rate of growth of the population decreases from the onset,an autocatalyti...This paper analyzes a population model with time-dependent advection and an autocatalytic-type growth.As opposed to a logistic growth where the rate of growth of the population decreases from the onset,an autocatalytic growth has a point of inflection where the rate of growth switches from an increasing trend to a decreasing trend.Employing the idea of Painleve property,we show that a variety of exact traveling wave solutions can be obtained for this model depending on the choice of the advection term.In particular,due to situations in resource distribution or environmental variations,if the advection is represented as a decaying function in time or an oscillating function in time,we are able to find exact solutions with interesting behavior.We also carry out a computational study of the model using an exponentially upwinded numerical scheme and illustrate the movement of the solutions and their characteristics pictorially.展开更多
文摘In this paper we discuss, an initial-boundary value problem of hyperbolic type with first derivative with respect to x. The asymptotic solution is constructed and its uniform validity is proved under weader compatibility conditions. Then we develop an exponentially fitted difference scheme and establish discrete energy inequality. Finally, we prove that the solution of difference problem uniformly converges to the solution of the original problem.
文摘In this article,we improve the order of precision of the two-dimensional Poisson equation by combining extrapolation techniques with high order schemes.The high order solutions obtained traditionally generate non-sparse matrices and the calculation time is very high.We can obtain sparse matrices by applying compact schemes.In this article,we compare compact and exponential finite difference schemes of fourth order.The numerical solutions are calculated in quadruple precision(Real*16 or extended precision)in FORTRAN language,and iteratively obtained until reaching the round-off error magnitude around 1.0E−32.This procedure is performed to ensure that there is no iteration error.The Repeated Richardson Extrapolation(RRE)method combines numerical solutions in different grids,determining higher orders of accuracy.The main contribution of this work is based on a process that initializes with fourth order solutions combining with RRE in order to find solutions of sixth,eighth,and tenth order of precision.The multigrid Full Approximation Scheme(FAS)is also applied to accelerate the convergence and obtain the numerical solutions on the fine grids.
基金supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University(IMSIU)(grant number IMSIU-DDRSP2503).
文摘This study develops a high-order computational scheme for analyzing unsteady tangent hyperbolic fluid flow with variable thermal conductivity,thermal radiation,and coupled heat andmass transfer effects.Amodified twostage Exponential Time Integrator is introduced for temporal discretization,providing second-order accuracy in time.A compact finite difference method is employed for spatial discretization,yielding sixth-order accuracy at most grid points.The proposed framework ensures numerical stability and convergence when solving stiff,nonlinear parabolic systems arising in fluid flow and heat transfer problems.The novelty of the work lies in combining exponential integrator schemes with compact high-order spatial discretization,enabling accurate and efficient simulations of tangent hyperbolic fluids under complex boundary conditions,such as oscillatory plates and varying thermal conductivity.This approach addresses limitations of classical Euler,Runge–Kutta,and spectral methods by significantly reducing numerical errors(up to 45%)and computational cost.Comprehensive parametric studies demonstrate how viscous dissipation,chemical reactions,the Weissenberg number,and the Hartmann number influence flow behaviour,heat transfer,and mass transfer.Notably,heat transfer rates increase by 18.6%with stronger viscous dissipation,while mass transfer rates rise by 21.3%with more intense chemical reactions.The real-world relevance of the study is underscored by its direct applications in polymer processing,heat exchanger design,radiative thermal management in aerospace,and biofluid transport in biomedical systems.The proposed scheme thus provides a robust numerical framework that not only advances the mathematical modelling of non-Newtonian fluid flows but also offers practical insights for engineering systems involving tangent hyperbolic fluids.
文摘Singularly perturbed boundary value problem with nonlocal conditions is examined. The appopriate solution exhibits boundary layer behavior for small positive values of the perturbative parameter. An exponentially fitted finite difference scheme on a non-equidistant mesh is constructed for solving this problem. The uniform convergence analysis in small parameter is given. Numerical example is provided, too.
文摘This paper applies exponentially fitted trapezoidal scheme to a stochastic oscillator. The scheme is convergent with mean-square order 1 and symplectic. Its numerical solution oscillates and the second moment increases linearly with time. The numerical example verifies the analysis of the scheme.
文摘An efficient and accurate exponential wave integrator Fourier pseudospectral （EWI-FP） method is proposed and analyzed for solving the symmetric regularized-long-wave （SRLW） equation, which is used for modeling the weakly nonlinear ion acoustic and space-charge waves. The numerical method here is based on a Gautschi-type exponential wave integrator for temporal approximation and the Fourier pseudospectral method for spatial discretization. The scheme is fully explicit and efficient due to the fast Fourier transform. Numerical analysis of the proposed EWI-FP method is carried out and rigorous error estimates are established without CFL-type condition by means of the mathematical induction. The error bound shows that EWI-FP has second order accuracy in time and spectral accuracy in space. Numerical results are reported to confirm the theoretical studies and indicate that the error bound here is optimal.
文摘This paper analyzes a population model with time-dependent advection and an autocatalytic-type growth.As opposed to a logistic growth where the rate of growth of the population decreases from the onset,an autocatalytic growth has a point of inflection where the rate of growth switches from an increasing trend to a decreasing trend.Employing the idea of Painleve property,we show that a variety of exact traveling wave solutions can be obtained for this model depending on the choice of the advection term.In particular,due to situations in resource distribution or environmental variations,if the advection is represented as a decaying function in time or an oscillating function in time,we are able to find exact solutions with interesting behavior.We also carry out a computational study of the model using an exponentially upwinded numerical scheme and illustrate the movement of the solutions and their characteristics pictorially.
