Department of Mathematics
Permanent URI for this collectionhttp://localhost:4000/handle/123456789/1920
Browse
Item Quasi-Monte Carlo algorithms for diffusion equations in high dimensions(Elsevier, 2005-02) Venkiteswaran, G.Diffusion equation posed on a high dimensional space may occur as a sub-problem in advection-diffusion problems (see [G. Venkiteswaran, M. Junk, A QMC approach for high dimensional Fokker–Planck equations modelling polymeric liquids, Math. Comput. Simul. 68 (2005) 43–56.] for a specific application). Although the transport part can be dealt with the method of characteristics, the efficient simulation of diffusion in high dimensions is a challenging task. The traditional Monte Carlo method (MC) applied to diffusion problems converges and is accurate, where N is the number of particles. It is well known that for integration, quasi-Monte Carlo (QMC) outperforms Monte Carlo in the sense that one can achieve convergence, up to a logarithmic factor. This is our starting point to develop methods based on Lécot’s approach [C. Lécot, F.E. Khettabi, Quasi-Monte Carlo simulation of diffusion, Journal of Complexity 15 (1999) 342–359.], which are applicable in high dimensions, with a hope to achieve better speed of convergence. Through a number of numerical experiments we observe that some of the QMC methods not only generalize to high dimensions but also show faster convergence in the results and thus, slightly outperform standard MC.Item A QMC approach for high dimensional Fokker–Planck equations modelling polymeric liquids(Elsevier, 2005-02) Venkiteswaran, G.A classical model used in the study of dynamics of polymeric liquids is the bead-spring chain representation of polymer molecules. The chain typically consists of a large number of beads and thus the state space of its configuration, which is essentially the position of all the constituent beads, turns out to be high dimensional. The distribution function governing the configuration of a bead-spring chain undergoing shear flow is a Fokker–Planck equation on . In this article, we present QMC methods for the approximate solution of the Fokker–Planck equation which are based on the time splitting technique to treat convection and diffusion separately. Convection is carried out by moving the particles along the characteristics and we apply the algorithms presented in [G. Venkiteswaran, M. Junk, QMC algorithms for diffusion equations in high dimensions, Math. Comput. Simul. 68 (2005) 23–41.] for diffusion. Altogether, we find that some of the QMC methods show reduced variance and thus slightly outperform standard MC.Item Quasi-Monte Carlo Simulation of Diffusion in a Spatially Nonhomogeneous Medium(Springer, 2009-11) Venkiteswaran, G.;We propose and test a quasi-Monte Carlo (QMC) method for solving the diffusion equation in the spatially nonhomogeneous case. For a constant diffusion coefficient, the Monte Carlo (MC) method is a valuable tool for simulating the equation: the solution is approximated by using particles and in every time step the displacement of each particle is drawn from a Gaussian distribution with constant variance. But for a spatially dependent diffusion coefficient, the straightforward extension using a spatially variable variance leads to biased results. A correction to the Gaussian steplength was recently proposed and provides satisfactory results. In the present work, we devise a QMC variant of this corrected MC scheme. We present the results of some numerical experiments showing that our QMC algorithm converges better than the corresponding MC method for the same number of particles.Item Quasi-Monte Carlo algorithms for diffusion equations in high dimensions(Elsevier, 2005-02) Venkiteswaran, G.Diffusion equation posed on a high dimensional space may occur as a sub-problem in advection-diffusion problems (see [G. Venkiteswaran, M. Junk, A QMC approach for high dimensional Fokker–Planck equations modelling polymeric liquids, Math. Comput. Simul. 68 (2005) 43–56.] for a specific application). Although the transport part can be dealt with the method of characteristics, the efficient simulation of diffusion in high dimensions is a challenging task. The traditional Monte Carlo method (MC) applied to diffusion problems converges and is accurate, where N is the number of particles. It is well known that for integration, quasi-Monte Carlo (QMC) outperforms Monte Carlo in the sense that one can achieve convergence, up to a logarithmic factor. This is our starting point to develop methods based on Lécot’s approach [C. Lécot, F.E. Khettabi, Quasi-Monte Carlo simulation of diffusion, Journal of Complexity 15 (1999) 342–359.], which are applicable in high dimensions, with a hope to achieve better speed of convergence. Through a number of numerical experiments we observe that some of the QMC methods not only generalize to high dimensions but also show faster convergence in the results and thus, slightly outperform standard MC.Item Extended Latin Hypercube Sampling for Integration and Simulation(Springer, 2013-01) Venkiteswaran, G.We analyze an extended form of Latin hypercube sampling technique that can be used for numerical quadrature and for Monte Carlo simulation. The technique utilizes random point sets with enhanced uniformity over the s-dimensional unit hypercube. A sample of N = n s points is generated in the hypercube. If we project the N points onto their ith coordinates, the resulting set of values forms a stratified sample from the unit interval, with one point in each subinterval [(k−1)/N,k/N). The scheme has the additional property that when we partition the hypercube into N subcubes ∏si=1[(ℓi−1)/n,ℓi/n), each one contains exactly one point. We establish an upper bound for the variance, when we approximate the volume of a subset of the hypercube, with a regular boundary. Numerical experiments assess that the bound is tight. It is possible to employ the extended Latin hypercube samples for Monte Carlo simulation. We focus on the random walk method for diffusion and we show that the variance is reduced when compared with classical random walk using ordinary pseudo-random numbers. The numerical comparisons include stratified sampling and Latin hypercube sampling.Item A robust numerical technique for weakly coupled system of parabolic singularly perturbed reaction–diffusion equations(Springer, 2023-02) Kumar, DevendraThis article presents a uniformly convergent numerical technique for a time-dependent reaction-dominated singularly perturbed system, including the same diffusion parameters multiplied with second-order spatial derivatives in all equations. Boundary layers are observed in the solution components for the small parameter. The proposed numerical technique consists of the Crank–Nicolson scheme in the temporal direction over a uniform mesh and quadratic B-splines collocation technique over an exponentially graded mesh in the spatial direction. We derived the robust error estimates to establish the optimal order of convergence. Numerical investigations confirm the theoretical determinations and the proposed method’s efficiency and accuracy.Item An efficient parameter uniform spline-based technique for singularly perturbed weakly coupled reaction-diffusion systems(Authero, 2022-07) Kumar, DevendraA parameter-uniform numerical scheme for a system of weakly coupled singularly perturbed reaction diffusion equations of arbitrary size with appropriate boundary conditions is investigated. More precisely, quadratic B-spline basis functions with an exponentially graded mesh are used to solve a ` × ` system whose solution exhibits parabolic (or exponential) boundary layers at both endpoints of the domain. A suitable mesh generating function is used to generate the exponentially graded mesh. The decomposition of the solution into regular and singular components is obtained to provide error estimates. A convergence analysis is addressed, which shows a uniform convergence of the second order. To validate the theoretical findings, two test problems are solved numericallyItem Numerical Simulation for Generalized Time-Fractional Burgers' Equation With Three Distinct Linearization Schemes(ASME, 2023-03) Kumar, DevendraIn the present study, we examined the effectiveness of three linearization approaches for solving the time-fractional generalized Burgers' equation using a modified version of the fractional derivative by adopting the Atangana-Baleanu Caputo derivative. A stability analysis of the linearized time-fractional Burgers' difference equation was also presented. All linearization strategies used to solve the proposed nonlinear problem are unconditionally stable. To support the theory, two numerical examples are considered. Furthermore, numerical results demonstrate the comparison of linearization strategies and manifest the effectiveness of the proposed numerical scheme in three distinct ways.Item A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation(De Gruyter, 2022-10) Kumar, DevendraIn this article, we present a novel numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ (BBMB) equation using Atangana Baleanu Caputo (ABC) derivative. First, we apply a linearization technique to deal with the generalized non-linear expression, and then the Crank–Nicolson finite difference formula is used in the temporal direction. A reliable numerical technique is applied to discretize the time-fractional ABC derivative, and the central difference formulae are used to approximate the derivatives in the spatial direction. The method is shown unconditionally stable and second-order convergent in both directions through the Fourier analysis. The numerical results of two test problems are analyzed to validate the theoretical results.Item Parameter independent scheme for singularly perturbed problems including a boundary turning point of multiplicity ≥ 1(JAAC, 2022) Kumar, DevendraA numerical scheme is developed for parabolic singularly perturbed boundary value problems, including multiple boundary turning points at the left endpoint of the spatial direction. The highest order derivative of these problems is multiplied by a small parameter , and when it is close to zero, the solution exhibits a parabolic type boundary layer near the left lateral surface of the domain of consideration. Thus, large oscillations appear when classical/standard numerical methods are used to solve the problem, and one cannot achieve the expected accuracy. Thus, the Crank-Nicolson scheme on a uniform mesh in the temporal direction and an upwind scheme on a Shishkin-type mesh in the spatial direction is constructed. The theoretical analysis shows that the method converges irrespective of the size of with accuracy . Three test examples are presented to verify that the computational results agree with the theoretical ones.Item Uniformly convergent scheme for fourth-order singularly perturbed convection-diffusion ODE(Elsevier, 2023-04) Kumar, DevendraThis paper contemplates a numerical investigation of the convection-diffusion type's fourth-order singularly perturbed linear and nonlinear boundary value problems. First, the considered linear fourth-order differential equation is converted into a strongly/weakly coupled singularly perturbed system (depending on the coefficient of the first-order derivative) of two ordinary differential equations with Dirichlet boundary conditions to solve the problem numerically. One of the equations is free from the perturbation parameter in the system. To obtain the solution for this system, we propose a numerical method of quadratic -splines on an exponentially graded mesh. Convergence analysis shows that the proposed numerical scheme is second-order uniformly convergent in the discrete maximum norm. The nonlinear differential equation is linearized using the quasilinearization technique, and then the proposed approach is applied to the linearized problem. The theoretical outcomes are validated by executing the proposed method on three test problems.Item A semi-analytic method for solving singularly perturbed twin-layer problems with a turning point(Vilnius Gediminas Technical University, 2023) Kumar, DevendraThis computational study investigates a class of singularly perturbed second-order boundary-value problems having dual (twin) boundary layers and simple turning points. It is well-known that the classical discretization methods fail to resolve sharp gradients arising in solving singularly perturbed differential equations as the perturbation (diffusion) parameter decreases, i.e., ε → 0+. To this end, this paper proposes a semi-analytic hybrid method consisting of a numerical procedure based on finite differences and an asymptotic method called the Successive Complementary Expansion Method (SCEM) to approximate the solution of such problems. Two numerical experiments are provided to demonstrate the method’s implementation and to evaluate its computational performance. Several comparisons with the numerical results existing in the literature are also made. The numerical observations reveal that the hybrid method leads to good solution profiles and achieves this in only a few iterations.Item Second-order convergent scheme for time-fractional partial differential equations with a delay in time(Springer, 2022-10) Kumar, DevendraThis paper aims to construct an effective numerical scheme to solve convection-reaction-diffusion problems consisting of time-fractional derivative and delay in time. First, the semi-discretization process is given for the fractional derivative using a finite-difference scheme with second-order accuracy. Then the cubic B-spline collocation method is employed to get the full discretization. We prove that the suggested scheme is conditionally stable and convergent. Two numerical examples are incorporated to verify the effectiveness of the algorithm. Numerical investigations support the proposed method’s accuracy and show that the method solves the problem efficiently.Item An efficient numerical technique for two-parameter singularly perturbed problems having discontinuity in convection coefficient and source term(Springer, 2023-01) Kumar, DevendraWe devise a spline-based numerical technique for a class of two-parameter singularly perturbed problems having discontinuous convection and source terms. The problem is discretized using the Crank–Nicolson formula in the temporal direction, and the trigonometric -spline basis functions are used in the spatial direction. The presence of perturbation parameters and the discontinuous convection/source terms result in the interior and boundary layers in the solution to the problem. Our primary focus is to resolve these layers and develop a uniformly convergent scheme. Initially, the proposed method gives almost first and second-order convergence in the spatial and temporal directions, respectively. Then, to improve the accuracy in the spatial direction, we have used the Richardson extrapolation technique. Two numerical examples are taken to demonstrate the layer phenomenon and confirm the theoretical proofs. It is evident from the tables that the Richardson extrapolation technique increases the accuracy from one to two in the spatial direction.Item A second-order numerical scheme for the time-fractional partial differential equations with a time delay(Springer, 2022-03) Kumar, DevendraThis work proposes a numerical scheme for a class of time-fractional convection–reaction–diffusion problems with a time lag. Time-fractional derivative is considered in the Caputo sense. The numerical scheme comprises the discretization technique given by Crank and Nicolson in the temporal direction and the spline functions with a tension factor are used in the spatial direction. Through the von Neumann stability analysis, the scheme is shown conditionally stable. Moreover, a rigorous convergence analysis is presented through the Fourier series. Two test problems are solved numerically to verify the effectiveness of the proposed numerical scheme.Item Wavelet-based approximation with nonstandard finite difference scheme for singularly perturbed partial integrodifferential equations(Springer, 2022-10) Kumar, DevendraA non-standard finite difference scheme with Haar wavelet basis functions is constructed for the convection–diffusion type singularly perturbed partial integrodifferential equations. The scheme comprises the Crank–Nicolson time semi-discretization followed by the Haar wavelet approximation in the spatial direction. The presence of the perturbation parameter leads to a boundary layer in the solution’s vicinity of x=1. The Shishkin mesh is constructed to resolve the boundary layer. The method is proved to be parameter-uniform convergent of order two in the L2-norm through meticulous error analysis. Compared to the recent methods developed to solve such problems, the present method is a boundary layer resolving, fast, and elegant.Item An effective numerical approach for two parameter time-delayed singularly perturbed problems(Springer, 2022-10) Kumar, DevendraA numerical scheme for the two-parameter singularly perturbed parabolic initial-boundary-value problems with a delay in time is considered. The solution to these problems exhibits twin boundary layers near the endpoints of the spatial domain. An appropriate piecewise-uniform mesh is constructed to resolve these layers. First, the given problem is semi-discretized in the temporal direction by employing the Crank–Nicolson scheme resulting in a system of ordinary differential equations at each time level. Then, to solve these systems, B-spline basis functions with the piecewise-uniform mesh leading to a tri-diagonal system of algebraic equations are used. The tri-diagonal system of algebraic equations is solved using the Thomas algorithm. Through rigorous analysis, we have shown that the scheme is second-order accurate in time and almost second-order accurate in space. Four test problems are solved to validate the theoretical results.Item Spline-based parameter-uniform scheme for fourth-order singularly perturbed differential equations(Springer, 2022-08) Kumar, DevendraThis paper considers a numerical study for the fourth-order singularly perturbed boundary value problems. The associated differential equation is converted into a weakly coupled system of two singularly perturbed ordinary differential equations with Dirichlet boundary conditions to solve the problem numerically. In the system, one of the equations is independent of the perturbation parameter. To solve this system, we present a numerical technique of quadratic B-splines on an exponentially graded mesh. The established results show that the scheme is second-order uniformly convergent in the discrete maximum norm. The theoretical results are validated using the proposed method on two test problems.Item Geometric mesh FDM for self-adjoint singular perturbation boundary value problems(Elsevier, 2007-07) Kumar, DevendraA numerical method based on finite difference method with variable mesh is given for second order singularly perturbed self-adjoint two point boundary value problems. The original problem is reduced to its normal form and the reduced problem is solved by FDM taking variable mesh(geometric mesh). The maximum absolute errors , for different values of parameter ϵ, number of points N, and the mesh ratio r, for three examples have been given in tables to support the efficiency of the method.Item Parameter-uniform fitted operator B-spline collocation method for self-adjoint singularly perturbed two-point boundary value problems(ETNA, 2008) Kumar, DevendraIn this paper, we develop a B-spline collocation method for the numerical solution of a self-adjoint singularly perturbed boundary value problem of the form We construct a fitting factor and use the B-spline collocation method, which leads to a tridiagonal linear system. The method is analyzed for parameter-uniform convergence. Several numerical examples are reported which demonstrate the efficiency of the proposed method.