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![1. The mean solution E[u(x)] and 100 sample paths obtained using the SFD method for the problem (4.1) using N = 8 (left) and N = 16 (right).](profile/Mahboub-Baccouch/publication/362068946/figure/fig1/AS:11431281153352097@1682424893261/The-mean-solution-Eux-and-100-sample-paths-obtained-using-the-SFD-method-for-the.jpg)
1. The mean solution E[u(x)] and 100 sample paths obtained using the SFD method for the problem (4.1) using N = 8 (left) and N = 16 (right).
![Fig. 4.1. The mean solution E[u(x)] and 100 sample paths obtained using...](profile/Mahboub-Baccouch/publication/362068946/figure/fig1/AS:11431281153352097@1682424893261/The-mean-solution-Eux-and-100-sample-paths-obtained-using-the-SFD-method-for-the_Q320.jpg)
![Fig. 4.3. The mean solution E[u(x)] and 100 sample paths obtained using...](profile/Mahboub-Baccouch/publication/362068946/figure/fig3/AS:11431281153329698@1682424893479/The-mean-solution-Eux-and-100-sample-paths-obtained-using-the-SFD-method-for_Q320.jpg)
![2: The errors maxi=0,1,...,N |E[u(xi)] − E[ui]| 2 , maxi=0,1,...,N...](profile/Mahboub-Baccouch/publication/362068946/figure/tbl1/AS:11431281153329699@1682424893541/2-The-errors-maxi0-1-N-Euxi-Eui-2-maxi0-1-N-Euxi-ui-2-and_Q320.jpg)
Citations
... Ramos and Rufai [9] implemented a third-derivative, two-step block Falkner approach to solve linear and non-linear BVPs. Some stochastic nonlinear second-order boundary value problems driven by additive noise has been solved by Baccouch [10] using the finite difference method. Numerous physical phenomena are modeled using strongly non-linear BVPs with specific parameter values. ...
In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results using the Galerkin weighted residual approach with Bernstein polynomials. The generation of residuals is brought on by the fact that our first approximation is computed using numerical methods. To minimize these residuals, we use the compact finite difference scheme of 4th-order convergence to solve the error differential equations in accordance with the error boundary conditions. We also introduce the formulation of the compact finite difference method of fourth-order convergence for the nonlinear BVPs. The improved approximations are produced by adding the error values derived from the approximations of the error differential equation to the weighted residual values. Numerical results are compared to the exact solutions and to the solutions available in the published literature to validate the proposed scheme, and high accuracy is achieved in all cases
... Therefore, the stochastic partial differential equation model has attracted the attention of many experts. For this type of model, there have been many effective numerical methods, such as stochastic Galerkin methods [19,20,21], stochastic configuration methods [22,23], sparse grid methods [24,25,26,27], perturbation methods [28,29] and many other methods [30,31,32,33,34,35,36]. The MCM is one of the most classical approaches for solving stochastic partial differential equations. ...
This paper develops a new 2D/3D stochastic closed-loop geothermal system with a random hydraulic conductivity tensor. We use the finite element method (FEM) and the Monte Carlo method (MCM) to discrete physical and probability spaces, respectively. This FEM-MCM method is effective. The stability for velocity and temperature is rigorously proved. Compared with the deterministic closed-loop geothermal system, a same optimal error estimate for approximate velocity and temperature is obtained. Furthermore, a series of numerical experiments were carried out to show this method has better stability and accuracy results.
... Up to now, different kinds of numerical methods have been applied to solving the form of the SPDE (1.1), such as finite difference methods, finite element methods, discontinuous Galerkin methods, WG methods, etc. In [2], the author presents a finite difference method for stochastic nonlinear secondorder boundary-value problems (BVPs) driven by additive noises, and proves that the finite difference solution converges to the solution to the original stochastic BVP at O(h) in the mean-square sense. The stochastic Allen-Cahn equation with additive noise is discretized by means of a spectral Galerkin method in space and a tamed version of the exponential Euler method in time [3]. ...
In this paper, a weak Galerkin (WG for short) finite element method is used to approximate nonlinear stochastic parabolic partial differential equations with spatiotemporal additive noises. We set up a semi-discrete WG scheme for the stochastic equations, and derive the optimal order for error estimates in the sense of strong convergence.
... and Dirichlet boundary conditions. The most physically and reasonable choice for the modeling of real-world problems is Neumann boundary conditions [5]. We consider the following general class of nonlinear singular, two-point boundary value problem (BVP): ...
... g(r, f (r)) = −f 5 (r) (5) then equation (1) arising in the study of equilibrium of isothermal gas sphere [1], [2], [8]. ...
... The comparison between our designed scheme and compact finite difference method (CFDM) proved that HHO-IPA is highly accurate. Table (5) confirms that the errors in HHO-IPA solution is smaller in magnitude than the CFDM [28]. Furthermore, the absolute error for m = 16, in HHO-IPA solution is 3.8619E − 08, however, for m = 16 CFDM reduced absolute error about 5.3227E − 07. ...
In the present research work, we designed a hybrid stochastic numerical solver to investigate nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions arising in various physical models. In this method, we hybridized Harris Hawks Optimizer with Interior Point Algorithm named HHO-IPA. We construct artificial neural networks (ANNs) model for the problem, and this model is tuned with the proposed scheme. This scheme overcomes the singular behavior of problems. The accuracy and applicability of the method are illustrated by finding absolute errors in the solution. The outcomes are compared with the results present in the literature to demonstrate the effectiveness and robustness of the scheme by considering four different nonlinear singular boundary value problems. Further, the convergence of the scheme is proved by performing computational complexity analysis. Moreover, the graphical overview of statistical analysis is added to our investigation to elaborate further on the scheme’s stability, accuracy, and consistency.
... , d − 1) denote mutually orthogonal unit tangential vectors to the interface Γ I , z is the hight, g is the gravitational acceleration, and Π(x) = K(x)ν g is the intrinsic permeability. The first interface condition (6) is governed by the conservation of mass, the second interface condition (7) represents the balance of the kinematic pressure in the matrix and the stress in the free flow at the normal direction along the interface, and the last interface condition (8) is the famous Beavers-Joseph condition [12,21,23,24,47,66,68,83,95,103]. ...
A multigrid multilevel Monte Carlo (MGMLMC) method is developed for the stochastic Stokes–Darcy interface model with random hydraulic conductivity both in the porous media domain and on the interface. Three interface conditions with randomness are considered on the interface between Stokes and Darcy equations, especially the Beavers–Joesph interface condition with random hydraulic conductivity. Because the randomness through the interface affects the flow in the Stokes domain, we investigate the coupled stochastic Stokes–Darcy model to improve the fidelity. Under suitable assumptions on the random coefficient, we prove the existence and uniqueness of the weak solution of the variational form. To construct the numerical method, we first adopt the Monte Carlo (MC) method and finite element method, for the discretization in the probability space and physical space, respectively. In order to improve the efficiency of the classical single-level Monte Carlo (SLMC) method, we adopt the multilevel Monte Carlo (MLMC) method to dramatically reduce the computational cost in the probability space. A strategy is developed to calculate the number of samples needed in MLMC method for the stochastic Stokes–Darcy model. In order to accomplish the strategy for MLMC method, we also present a practical method to determine the variance convergence rate for the stochastic Stokes–Darcy model with Beavers–Joseph interface condition. Furthermore, MLMC method naturally provides the hierarchical grids and sufficient information on these grids for multigrid (MG) method, which can in turn improve the efficiency of MLMC method. In order to fully make use of the dynamical interaction between this two methods, we propose a multigrid multilevel Monte Carlo (MGMLMC) method with finite element discretization for more efficiently solving the stochastic model, while additional attention is paid to the interface and the random Beavers–Joesph interface condition. The computational cost of the proposed MGMLMC method is rigorously analyzed and compared with the SLMC method. Numerical examples are provided to verify and illustrate the proposed method and the theoretical conclusions.
... A number of efficient numerical methods have been developed to solve stochastic PDEs, such as polynomial chaos [69,116,117], stochastic Galerkin method [8,35,86,98], stochastic collocation method [7,59], sparse grid methods [11,87,90,91], multilevel Monte Carlo method [14,29,42,52,73,97,104], and many others [9,12,27,82,103,107,109,112,114,115,118,121,122]. These methods have also been applied to solve the stochastic optimization and control problems [4,13,34,58,105]. ...
In this paper, we develop a sparse grid stochastic collocation method to improve the computational efficiency in handling the steady Stokes-Darcy model with random hydraulic conductivity. To represent the random hydraulic conductivity, the truncated Karhunen-Loève expansion is used. For the discrete form in probability space, we adopt the stochastic collocation method and then use the Smolyak sparse grid method to improve the efficiency. For the uncoupled deterministic subproblems at collocation nodes, we apply the general coupled finite element method. Numerical experiment results are presented to illustrate the features of this method, such as the sample size, convergence, and randomness transmission through the interface.
In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results using the Galerkin weighted residual approach with Bernstein polynomials. The generation of residuals is brought on by the fact that our first approximation is computed using numerical methods. To minimize these residuals, we use the compact finite difference scheme of 4th-order convergence to solve the error differential equations in accordance with the error boundary conditions. We also introduce the formulation of the compact finite difference method of fourth-order convergence for the nonlinear BVPs. The improved approximations are produced by adding the error values derived from the approximations of the error differential equation to the weighted residual values. Numerical results are compared to the exact solutions and to the solutions available in the published literature to validate the proposed scheme, and high accuracy is achieved in all cases.
