Showing papers on "Finite difference published in 2021"

Solving nonlinear differential equations with differentiable quantum circuits

Abstract: We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. We simulate the algorithm to solve an instance of Navier-Stokes equations and compute density, temperature, and velocity profiles for the fluid flow in a convergent-divergent nozzle.

A nonstandard finite difference scheme for the modeling and nonidentical synchronization of a novel fractional chaotic system

Abstract: The aim of this paper is to introduce and analyze a novel fractional chaotic system including quadratic and cubic nonlinearities. We take into account the Caputo derivative for the fractional model and study the stability of the equilibrium points by the fractional Routh–Hurwitz criteria. We also utilize an efficient nonstandard finite difference (NSFD) scheme to implement the new model and investigate its chaotic behavior in both time-domain and phase-plane. According to the obtained results, we find that the new model portrays both chaotic and nonchaotic behaviors for different values of the fractional order, so that the lowest order in which the system remains chaotic is found via the numerical simulations. Afterward, a nonidentical synchronization is applied between the presented model and the fractional Volta equations using an active control technique. The numerical simulations of the master, the slave, and the error dynamics using the NSFD scheme are plotted showing that the synchronization is achieved properly, an outcome which confirms the effectiveness of the proposed active control strategy.

A Theoretical and Empirical Comparison of Gradient Approximations in Derivative-Free Optimization

Abstract: In this paper, we analyze several methods for approximating gradients of noisy functions using only function values. These methods include finite differences, linear interpolation, Gaussian smoothing, and smoothing on a sphere. The methods differ in the number of functions sampled, the choice of the sample points, and the way in which the gradient approximations are derived. For each method, we derive bounds on the number of samples and the sampling radius which guarantee favorable convergence properties for a line search or fixed step size descent method. To this end, we use the results in Berahas et al. (Global convergence rate analysis of a generic line search algorithm with noise, arXiv:1910.04055 , 2019) and show how each method can satisfy the sufficient conditions, possibly only with some sufficiently large probability at each iteration, as happens to be the case with Gaussian smoothing and smoothing on a sphere. Finally, we present numerical results evaluating the quality of the gradient approximations as well as their performance in conjunction with a line search derivative-free optimization algorithm.

A Positivity-Preserving, Energy Stable And Convergent Numerical Scheme For The Poisson-Nernst-Planck System

Abstract: In this paper we propose and analyze a finite difference numerical scheme for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility $H^{-1}$ gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivity-preserving property for both concentrations, $n$ and $p$, is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of $0$ prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish the $\ell^\infty$ bound for $n$ and $p$), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this article, which demonstrates the robustness of the proposed numerical scheme.

Meshless upwind local radial basis function-finite difference technique to simulate the time- fractional distributed-order advection–diffusion equation

Abstract: The main objective in this paper is to propose an efficient numerical formulation for solving the time-fractional distributed-order advection–diffusion equation. First, the distributed-order term has been approximated by the Gauss quadrature rule. In the next, a finite difference approach is applied to approximate the temporal variable with convergence order $$\mathcal{O}(\tau ^{2-\alpha })$$ as $$0<\alpha <1$$ . Finally, to discrete the spacial dimension, an upwind local radial basis function-finite difference idea has been employed. In the numerical investigation, the effect of the advection coefficient has been studied in terms of accuracy and stability of the proposed difference scheme. At the end, two examples are studied to approve the impact and ability of the numerical procedure.

Effect of thermal radiation on conjugate natural convection flow of a micropolar fluid along a vertical surface

Abstract: This article investigates the behavior of conjugate natural convection over a finite vertical surface immersed in a micropolar fluid in the presence of intense thermal radiation. The governing boundary layer equations are made dimensionless and then transformed into suitable form by introducing the non-similarity transformations. The reduced system of parabolic partial differential equations is integrated numerically along the vertical plate by using an implicit finite difference Keller-box method. The features of fluid flow and heat transfer characteristics for various values of micropolar or material parameter, K , conjugate parameter, B , and thermal radiation parameter, R d , are analyzed and presented graphically. Results are presented for the local skin friction coefficient, heat transfer rate and couple stress coefficient for high Prandtl number. It is found that skin friction coefficient and couple stress coefficient reduces whereas heat transfer rate enhances when the micro-inertia parameter increases. All the physical quantities get augmented with thermal radiation.

Numerical solution of the fractional Rayleigh–Stokes model arising in a heated generalized second-grade fluid

Abstract: This paper addresses the solution of the Rayleigh–Stokes problem for an edge in a generalized Oldroyd-B fluid using fractional derivatives and the radial basis function-generated finite difference (RBF-FD) method. The time discretization is accomplished via the finite difference approach, while the spatial derivative terms are discretized using the local RBF-FD. The main idea is to consider the distribution of the data nodes within the local support domain so that the number of nodes remains constant. In addition, the stability and convergence analysis of the proposed method are discussed. The results using the RBF-FD are compared with those of other techniques on irregular domains showing the feasibility and efficiency of the new approach.

Numerical solution of two-dimensional stochastic time-fractional Sine–Gordon equation on non-rectangular domains using finite difference and meshfree methods

Abstract: The nonlinear Sine-Gordon equation is one of the widely used partial differential equations that appears in various sciences and engineering. The main purpose of writing this article is providing an efficient numerical method for solving two-dimensional (2D) time-fractional stochastic Sine–Gordon equation on non-rectangular domains. In this method, radial basis functions (RBFs) and finite difference scheme are used to calculate the approximate solution of the mentioned problem. The complexity of solving this problem arises from its high dimension, irregular area, stochastic and fractional terms. Finite difference technique is applied to overcome on the problem dimension, whereas interpolation method based on RBFs is the best idea for solving problems defined in irregular domains. The stochastic Sine–Gordon equation is transformed into elliptic stochastic differential equations using the finite difference method and meshfree method based on RBFs are used to approximate the obtained stochastic differential equation. Some numerical examples are included to investigate the efficiency and accuracy of the presented method.

A computational approach for the space-time fractional advection–diffusion equation arising in contaminant transport through porous media

Abstract: The fractional advection–diffusion equation, known as non-local diffusion, is a relationship utilized in groundwater hydrology as a reliable means of modeling the transport of passive tracers in porous media by fluid flow. The main target of this paper is to develop a numerical method for solving the space-time fractional advection–diffusion equation (STFADE) defined by Caputo sense. In this way, a finite difference formula with first-order accuracy is used to discretize the problem in the temporal direction, and also the Chebyshev collocation method of the third kind is applied to approximate the space variable. The stability and convergence of the fully discrete scheme are rigorously established in $$L^{2}$$ norm. The numerical results are presented and are compared with other methods to show the capability of the numerical scheme proposed here.

A local radial basis function-finite difference (RBF-FD) method for solving 1D and 2D coupled Schrödinger-Boussinesq (SBq) equations

Abstract: In this study, one-dimensional (1D) and two-dimensional (2D) coupled Schrodinger-Boussinesq (SBq) equations are examined numerically. A local meshless method based on radial basis function-finite difference (RBF-FD) method for spatial approximation is devised. We use polyharmonic splines as radial basis function along with augmented polynomials. By using polyharmonic splines we avoid to choose optimal shape parameter which requires special algorithms in meshless methods. For temporal discretization, low-storage ten-stage fourth-order explicit strong stability preserving Runge Kutta method is used which gives more flexibility on temporal step width. L ∞ and L 2 error norms are calculated to show accuracy of the proposed method. Further, conserved quantities are monitoried during numerical simulations to see how good the proposed method preserves them. Stability of the proposed method is dicussed numerically. Some codes are developed in Julia programming language to achieve more speed up in numerical simulations. Obtained results and their comparison with some studies such as wavelet, difference schemes and Fourier spectral methods available in literature verify the efficiency and reliability of the proposed method.

Finite Difference and Spline Approximation for Solving Fractional Stochastic Advection-Diffusion Equation

Abstract: This paper is concerned with numerical solution of time fractional stochastic advection-diffusion type equation where the first order derivative is substituted by a Caputo fractional derivative of order $$\alpha $$ ( $$0 <\alpha \le 1$$ ). This type of equations due to randomness can rarely be solved, exactly. In this paper, a new approach based on finite difference method and spline approximation is employed to solve time fractional stochastic advection-diffusion type equation, numerically. After implementation of proposed method, the under consideration equation is transformed to a system of second order differential equations with appropriate boundary conditions. Then, using a suitable numerical method such as the backward differentiation formula, the resulting system can be solved. In addition, the error analysis is shown in some mild conditions by ignoring the error terms $$O(\Delta t^2)$$ in the system. In order to show the pertinent features of the suggested algorithm such as accuracy, efficiency and reliability, some test problems are included. Comparison achieved results via proposed scheme in the case of classical stochastic advection-diffusion equation ( $$\alpha =1$$ ) with obtained results via wavelets Galerkin method and obtained results for other values of $$\alpha $$ with the values of exact solution confirm the validity, efficiency and applicability of the proposed method.

Mesh sampling and weighting for the hyperreduction of nonlinear Petrov–Galerkin reduced-order models with local reduced-order bases

Abstract: The energy-conserving sampling and weighting (ECSW) method is a hyperreduction method originally developed for accelerating the performance of Galerkin projection-based reduced-order models (PROMs) associated with large-scale finite element models, when the underlying projected operators need to be frequently recomputed as in parametric and/or nonlinear problems. In this paper, this hyperreduction method is extended to Petrov-Galerkin PROMs where the underlying high-dimensional models can be associated with arbitrary finite element, finite volume, and finite difference semi-discretization methods. Its scope is also extended to cover local PROMs based on piecewise-affine approximation subspaces, such as those designed for mitigating the Kolmogorov $n$-width barrier issue associated with convection-dominated flow problems. The resulting ECSW method is shown in this paper to be robust and accurate. In particular, its offline phase is shown to be fast and parallelizable, and the potential of its online phase for large-scale applications of industrial relevance is demonstrated for turbulent flow problems with $O(10^7)$ and $O(10^8)$ degrees of freedom. For such problems, the online part of the ECSW method proposed in this paper for Petrov-Galerkin PROMs is shown to enable wall-clock time and CPU time speedup factors of several orders of magnitude while delivering exceptional accuracy.

Mean field games of controls: Finite difference approximations

Abstract: We consider a class of mean field games in which the agents interact through both their states and controls, and we focus on situations in which a generic agent tries to adjust her speed (control) to an average speed (the average is made in a neighborhood in the state space). In such cases, the monotonicity assumptions that are frequently made in the theory of mean field games do not hold, and uniqueness cannot be expected in general. Such model lead to systems of forward-backward nonlinear nonlocal parabolic equations; the latter are supplemented with various kinds of boundary conditions, in particular Neumann-like boundary conditions stemming from reflection conditions on the underlying controled stochastic processes. The present work deals with numerical approximations of the above megntioned systems. After describing the finite difference scheme, we propose an iterative method for solving the systems of nonlinear equations that arise in the discrete setting; it combines a continuation method, Newton iterations and inner loops of a bigradient like solver. The numerical method is used for simulating two examples. We also make experiments on the behaviour of the iterative algorithm when the parameters of the model vary.

Convergence analysis of an L1-continuous Galerkin method for nonlinear time-space fractional Schrödinger equations

Abstract: This paper develops and analyses a finite difference/spectral-Galerkin scheme for the nonlinear fractional Schrodinger equations with Riesz space- and Caputo time-fractional derivatives. The L 1 fi...

Combined Galerkin spectral/finite difference method over graded meshes for the generalized nonlinear fractional Schrödinger equation

Abstract: The main aim of this paper is to construct an efficient Galerkin–Legendre spectral approximation combined with a finite difference formula of L1 type to numerically solve the generalized nonlinear fractional Schrodinger equation with both space- and time-fractional derivatives. We discretize the Riesz space-fractional derivative using the Legendre–Galerkin spectral method and the time-fractional derivative using the L1 scheme on nonuniform meshes. The stability and convergence analyses of the numerical scheme are studied in detail. The scheme is unconditionally stable and convergent of $$\min \{\kappa \theta ,2-\theta \}$$ order convergence in time and of spectral accuracy in space, where $$\theta $$ is the order of fractional derivative and $$\kappa $$ is the grading mesh parameter. To verify the efficiency of the proposed algorithm, two numerical test problems are performed with convergence and error analysis.

A finite-difference procedure to solve weakly singular integro partial differential equation with space-time fractional derivatives

Abstract: The main aim of the current paper is to propose an efficient numerical technique for solving space-time fractional partial weakly singular integro-differential equation. The temporal variable is based on the Riemann–Liouville fractional derivative and the spatial direction is based on the Riesz fractional derivative. Thus, to achieve a numerical technique, the time variable is discretized using a finite difference scheme with convergence order $${{\mathcal {O}}}(\tau ^{\frac{3}{2}})$$ . Also, the space variable is discretized using a finite difference scheme with second-order accuracy. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, two test problems have been illustrated to verify the efficiency, applicability and simplicity of the proposed technique.