Equidistribution Principle Research Articles

Adaptive mesh based efficient approximations for Darcy scale precipitation–dissolution models in porous media

AbstractIn this work, we consider the Darcy scale precipitation–dissolution reactive transport 1D and 2D models in a porous medium and provide the adaptive mesh based numerical approximations for solving them efficiently. These models consist of a convection‐diffusion‐reaction PDE with reactions being described by an ODE having a nonlinear, discontinuous, possibly multi‐valued right hand side describing precipitate concentration. The bulk concentration in the aqueous phase develops fronts and the precipitate concentration is described by a free and time‐dependent moving boundary. The time adaptive moving mesh strategy, based on equidistribution principle in space and governed by a moving mesh PDE, is utilized and modified in the context of present problem for finite difference set up in 1D and finite element set up in 2D. Moreover, we use a predictor corrector based algorithm to solve the nonlinear precipitation–dissolution models. For equidistribution approach, we choose an adaptive monitor function and smooth it based on a diffusive mechanism. Numerical tests are performed to demonstrate the accuracy and efficiency of the proposed method by examples through finite difference approach for 1D and finite element approach in 2D. The moving mesh refinement accurately resolves the front location of Darcy scale precipitation–dissolution reactive transport model and reduces the computational cost in comparison to numerical simulations using a fixed mesh.

In the Quest for Scale-optimal Mappings

Optimal mapping is one of the longest-standing problems in computational mathematics. It is natural to measure the relative curve length error under map to assess its quality. The maximum of such error is called the quasi-isometry constant, and its minimization is a nontrivial max-norm optimization problem. We present a physics-based quasi-isometric stiffening (QIS) algorithm for the max-norm minimization of hyperelastic distortion. QIS perfectly equidistributes distortion over the entire domain for the ground-truth test (unit hemisphere flattening) and, when it is not possible, tends to create zones where all cells have the same distortion. Such zones correspond to fragments of elastic material that became rigid under stiffening, reaching the deformation limit. As such, maps built by QIS are related to the de Boor equidistribution principle, which asks for an integral of a certain error indicator function to be the same over each mesh cell. Under certain assumptions on the minimization toolbox, we prove that our method can build, in a finite number of steps, a deformation whose maximum distortion is arbitrarily close to the (unknown) minimum. We performed extensive testing: on more than 10,000 domains QIS was reliably better than the competing methods. In summary, we reliably build 2D and 3D mesh deformations with the smallest known distortion estimates for very stiff problems.

Parameter uniform higher order numerical treatment for singularly perturbed Robin type parabolic reaction diffusion multiple scale problems with large delay in time

In this paper, we address a class of boundary layer originated singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions having large time delay; for the higher order numerical analysis. The boundary layer originated nonuniform mesh, is generated by adaptive moving mesh approach based on equidistribution principle. The mesh is obtained by solving a nonlinear system; involving the discrete problem and the discrete nonlinear mesh generating functions. We provide the theoretical mesh structure. The difference scheme for Robin type boundary conditions having normalized flux; uses a combination of space and time discretizations so that the order of convergence in space is higher than the usual linear accurate upwind discretization in space. The present theoretical convergence analysis requires less regularity of the solution and proves to be exactly second order parameter uniform accurate in space. Numerical experiments highly validate the theoretical findings and also show that the present approach is better than the well-known upwind discretizations.

USING PINN TO SOLVE EQUATIONS OF EQUIDISTRIBUTIONAL METHOD FOR CONSTRUCTING 2D STRUCTURED ADAPTED NUMERICAL GRIDS

The Equidistributional method is a popular technique for constructing numerical grids in engineering and scientific simulations. It is based on the principle of equidistribution, which requires evenly spaced grid points to reduce numerical errors. However, traditional Equidistributional methods can become inefficient and inaccurate for complex geometries and boundary conditions. In this paper, we present a new approach for solving the Equidistributional method's equations using physics-informed neural networks (PINN). PINN is a type of machine learning algorithm that has been shown to be effective for solving partial differential equations (PDEs). Our findings suggest that the use of PINN has the potential to significantly enhance the performance of the Equidistributional method for constructing 2D structured adapted numerical grids.         TRANSLATE with x English Arabic Hebrew Polish Bulgarian Hindi Portuguese Catalan Hmong Daw Romanian Chinese Simplified Hungarian Russian Chinese Traditional Indonesian Slovak Czech Italian Slovenian Danish Japanese Spanish Dutch Klingon Swedish English Korean Thai Estonian Latvian Turkish Finnish Lithuanian Ukrainian French Malay Urdu German Maltese Vietnamese Greek Norwegian Welsh Haitian Creole Persian   //   TRANSLATE with COPY THE URL BELOW Back EMBED THE SNIPPET BELOW IN YOUR SITE Enable collaborative features and customize widget: Bing Webmaster Portal Back //     Язык этой страницы: Английский   Перевести на Русский         Азербайджанский Албанский Амхарский Английский Арабский Армянский Африкаанс Бенгальский Бирманский Болгарский Валлийский Венгерский Вьетнамский Греческий Гуджарати Датский Иврит Индонезийский Исландский Испанский Итальянский Казахский Каннада Каталанский Китайский (традиционный) Китайский (упрощенный) Корейский Креольский (гаити) Курманджи Кхмерский Лаосский Латышский Литовский Малагасийский Малайский Малаялам Мальтийский Маори Маратхи Немецкий Непальский Нидерландский Норвежский Панджаби Персидский Польский Португальский Пушту Румынский Русский Самоанский Словацкий Словенский Тайский Тамильский Телугу Турецкий Украинский Урду Финский Французский Хинди Хорватский Чешский Шведский Эстонский Японский   Всегда переводить Английский на РусскийPRO Никогда не переводить Английский Никогда не переводить jpcsip.kaznu.kz

A robust adaptive grid method for first-order nonlinear singularly perturbed Fredholm integro-differential equations

<abstract><p>In this paper, a robust adaptive grid method is developed for solving first-order nonlinear singularly perturbed Fredholm integro-differential equations (SPFIDEs). Firstly such SPFIDEs are discretized by the backward Euler formula for differential part and the composite numerical quadrature rule for integral part. Then both a prior and an a posterior error analysis in the maximum norm are derived. Based on the prior error bound and the mesh equidistribution principle, it is proved that there exists a mesh gives optimal first-order convergence which is robust with respect to the perturbation parameter. Finally, the posterior error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Numerical results are given to illustrate our theoretical result.</p></abstract>

An adaptive grid method for a singularly perturbed convection-diffusion equation with a discontinuous convection coefficient

<abstract><p>In this paper, an adaptive grid method is put forward to solve a singularly perturbed convection-diffusion problem with a discontinuous convection coefficient. First, this problem is discretized by using an upwind finite difference scheme on an arbitrary nonuniform grid except the fixed jump point. Then, a first-order maximum norm a posterior error estimate is derived. Further, based on this a posteriori error estimation and the mesh equidistribution principle, an adaptive grid generation algorithm is constructed. Finally, some numerical experiments are presented that support our theoretical estimate.</p></abstract>

An Optimal Adaptive Grid Method Based on L1 Scheme for a Nonlinear Caputo Fractional Differential Equation

A nonlinear fractional differential equation with a Caputo derivative of order α is studied. This problem is discretized by using the L1 scheme on an arbitrary nonuniform mesh. By utilizing the Taylor expansion with integral remainder term, an optimal local truncation error estimation of L1 scheme is proved. Based on this truncation error estimation and the mesh equidistribution principle, a new monitor function is constructed to construct an adaptive grid generation algorithm. Numerical experiments are performed to confirm the accuracy of our new adaptive grid algorithm.

A Second-Order Adaptive Grid Method for a Singularly Perturbed Volterra Integrodifferential Equation

In this paper, an adaptive grid method for a singularly perturbed Volterra integro-differential equation is studied. Firstly, this problem is discretized by a new second-order finite difference scheme, for which a truncation error analysis is conducted. Then, based on this truncation error bound and the mesh equidistribution principle, we show that there is a mesh that provides an optimal error bound of O(N−2), which is robust with respect to the perturbation parameter. Finally, based on an approximation monitor function, an adaptive grid generation algorithm is constructed and some numerical results are given to support our theoretical results.

A mesh adaptation algorithm using new monitor and estimator function for discontinuous and layered solution

<p style='text-indent:20px;'>In this work a novel mesh adaptation technique is proposed to approximate discontinuous or boundary layer solution of partial differential equations. We introduce new estimator and monitor function to detect solution region containing discontinuity and layered region. Subsequently, this information is utilized along with equi-distribution principle to adapt the mesh locally. Numerical tests for numerous scalar problems are presented. These results clearly demonstrate the robustness of this method and non-oscillatory nature of the computed solutions.</p>

A Novel Second-Order Adaptive Grid Method for Singularly Perturbed Convection–Diffusion Equations

In this paper, a singularly perturbed convection–diffusion equation is studied. At first, the original problem is transformed into a parameterized singularly perturbed Volterra integro-differential equation by using an integral transform. Then, a second-order finite difference method on an arbitrary mesh is given. The stability and local truncation error estimates of the discrete schemes are analyzed. Based on the mesh equidistribution principle and local truncation error estimation, an adaptive grid algorithm is given. In addition, in order to calculate the parameters of the transformation equation, a nonlinear unconstrained optimization problem is constructed. Numerical experiments are given to illustrate the effectiveness of our presented adaptive grid algorithm.

Numerical analysis of a system of semilinear singularly perturbed first‐order differential equations on an adaptive grid

An adaptive grid method based on the backward Euler formula for a system of semilinear singularly perturbed initial value problems is studied. Based on the a priori error analysis and mesh equidistribution principle, we prove that the convergence rate of our semidiscrete adaptive grid method is first order, which is robust with respect to the perturbation parameters. Then, in order to construct a fully discrete adaptive grid method, a standard residual‐type a posterior error estimation is constructed by using the linear polynomial interpolation technique. A partly heuristic argument based on this a posteriori error estimator leads to an optimal monitor function, which is used to design an adaptive grid algorithm. Furthermore, we also extend our presented adaptive grid method to a nonlinear system of singularly perturbed problem arising in the modeling of enzyme kinetics and a system of singularly perturbed delay differential equations, respectively. Finally, numerical results are provided to illustrate the effectiveness of our presented adaptive grid method.

Katmanlardaki Asal Sayılar

In this article, unlike the known deterministic and probabilistic methods used in determining prime numbers, a new type of deterministic method based on only algebraic analysis will be proven without creating a designed equation and without writing preliminary assumptions and prerequisites. For this, in the layers of the numbers in mod 30, it will be proven that the places 30k+1, 30k+7, 30k+11, 30k+13, 30k+17, 30k+19, 30k+23, 30k+29 where the prime numbers except for the first three locate, form a closed system in themselves. These eight locations will be referred to as eight layers and it will also be explained that the prime numbers are distributed equally across eight layers (equidistribution principle) in the closed system. With this new method, it will also be shown that there are possibilities to reduce the processing load considerably.

Richardson Extrapolation Method on an Adaptive Grid for Singularly Perturbed Volterra Integro-Differential Equations

In this paper, an adaptive grid method for singularly perturbed Volterra integro-differential equations exhibiting a boundary layer is studied. At first, based on an arbitrary nonuniform mesh, we use the classical backward Euler formula to discrete the first-order derivative part and the left rectangle formula to approximate the integral part, respectively. Meanwhile, it is shown from the mesh equidistribution principle and the truncation error analysis that the presented adaptive grid method is first-order uniformly convergent with respect to the perturbation parameter. Furthermore, the Richardson extrapolation technique is utilized to improve the uniform accuracy of the presented adaptive grid method in the discrete supremum norm form to where is the mesh parameter. Finally, some numerical results are given to support the theoretical results.

Analysis of a finite difference scheme for a nonlinear Caputo fractional differential equation on an adaptive grid

<abstract> A nonlinear initial value problem whose differential operator is a Caputo derivative of order $ \alpha $ with $ 0 &lt; \alpha &lt; 1 $ is studied. By using the Riemann-Liouville fractional integral transformation, this problem is reformulated as a Volterra integral equation, which is discretized by using the right rectangle formula. Both a priori and an a posteriori error analysis are conducted. Based on the a priori error bound and mesh equidistribution principle, we show that there exists a nonuniform grid that gives first-order convergent result, which is robust with respect to $ \alpha $. Then an a posteriori error estimation is derived and used to design an adaptive grid generation algorithm. Numerical results complement the theoretical findings. </abstract>

Convergence analysis of Richardson extrapolation for a quasilinear singularly perturbed problem with an integral boundary condition on an adaptive grid

The numerical solution of a quasilinear singularly perturbed problem with an integral boundary condition is considered. For discretizing the differential equation, we use the backward Euler formula and for the integral boundary condition the left rectangle formula is constructed on an arbitrary nonuniform mesh. Based on the truncation error analysis and mesh equidistribution principle, a suitable monitor function is chosen to obtain an adaptive grid. Furthermore, the convergence analysis of our discretization scheme on an adaptive grid before and after Richardson extrapolation is carried out, which is shown that the use of Richardson extrapolation improves the uniform accuracy in the discrete maximum norm from first-order to second-order. Finally, some numerical results are given to illustrate the theoretical results.

High‐order convergent methods for singularly perturbed quasilinear problems with integral boundary conditions

In this work, we develop a numerical scheme for a class of singularly perturbed quasilinear problems with integral boundary conditions. The quasilinear equation is discretized using a hybrid scheme, and the composite trapezoidal rule is used to discretize the boundary condition. We construct a general error analysis framework for the discrete scheme. Within this framework, the discrete scheme is shown to be uniformly convergent of on Shishkin meshes and on Bakhvalov meshes. Further, we propose adaptive generation of meshes based on a suitable monitor function and the mesh equidistribution principle. We prove that on these meshes the discrete scheme is uniformly convergent of Our theoretical findings are supported by numerical results obtained through experiments.

SDFEM for singularly perturbed parabolic initial-boundary-value problems on equidistributed grids

In this article, we study the convergence properties of the streamline-diffusion finite element method (SDFEM) for singularly perturbed 1D parabolic convection–diffusion initial-boundary-value problems. To discretize the spatial domain, we use a layer-adaptive nonuniform grids obtained through the equidistribution principle, whereas uniform grid is used in the time direction. Here, we use the backward-Euler method to discretize the temporal derivative and the SDFEM scheme for the spatial derivatives. The proposed method is uniformly convergent with first-order in time and second-order in space.

Finite element modelling of geophysical electromagnetic data with goal-oriented hr-adaptivity

Large discontinuities (jumps) in coefficients often appear when modelling physical problems involving inhomogeneous media. An example of this is the geophysical electromagnetic (EM) problem, where these jumps occur at interfaces which separate regions of (high) conductivity contrasts. These interfaces, along with other problem features, such as singular EM sources, motivate the use of adaptive mesh refinement to improve the efficiency of the solution of forward problems. Also, pointwise observations are made in geophysical EM surveys, which motivates the use of goal-oriented mesh adaptivity. In this work, we study the combined application of h- and r-refinements for the modelling of geophysical EM data. The proposed hr-adaptivity algorithm is thoroughly investigated in 1D, and then extended, and numerically validated, in 2D, using simple examples as well as a realistic model with irregular interfaces. In 1D, the steady-state diffusion and Helmholtz equations, which are commonly solved for the EM scalar and vector potentials, respectively [16], constitute the physical partial differential equations (PPDEs). In 2D, the Helmholtz equation is used as the PPDE. Additionally, a real 2D problem with a benchmark model is considered where the transverse electric (TE) mode of Maxwell’s equations is used as the PPDE. The r-refinements are based on an equidistribution principle and variational methods, for the 1D and 2D cases, respectively, which lead to mesh PDEs (MPDEs). The coupled PPDE and MPDE are solved in an iterative manner to enhance the accuracy of the PPDE solution by improving the equidistribution of a monitor function based on a posteriori error estimates. The results, in 1D, with two hierarchical error–based monitor functions and a residual error–based monitor function display the similarity between these functions in terms of the accuracy that could be achieved. In both 1D and 2D, comparisons are made between global and goal-oriented adaptivity which show the advantage of goal-oriented error estimates in gaining higher accuracy at a target, compared to global error estimates. In 2D, the results also demonstrate the higher efficiency of hr-adaptivity compared with pure h-refinement, in terms of computation time.

Numerical Solution of Quasilinear Singularly Perturbed Problems by the Principle of Equidistribution

In this paper, the numerical solution and its error analysis of quasilinear singular perturbation two-point boundary value problems based on the principle of equidistribution are given. On the non-uniform grid of the uniformly distributed arc-length monitor function, the solution of the simple upwind scheme is obtained. It is proved that the adaptive simple upwind scheme based on the principle of equidistribution has uniform convergence for small perturbation parameters. Numerical experiments are carried out and the error analysis are confirmed.

Higher Order Non-Uniform Grids for Singularly Perturbed Convection-Diffusion-Reaction Problems

In this paper, a higher order nonuniform grid strategy is developed for solving singularly perturbed convection-diffusion-reaction problems with boundary layers. A new nonuniform grid finite difference method (FDM) based on a coordinate transformation is adopted to establish higher order accuracy. To achieve this, we study and make use of the truncation error of the discretized system to obtain a fourth-order nonuniform grid transformation. Considering a three-point central finite-difference scheme, we create not only fourth-order but even sixth-order approximations (which is the maximum order that can be obtained) by a suitable choice of the underlying nonuniform grids. Further, an adaptive nonuniform grid method based on equidistribution principle is used to demonstrate the sixth-order of convergence. Unlike several other adaptive numerical methods, our strategy uses no pre-knowledge of the location and the width of the layers. Numerical experiments for various test problems are presented to verify the theoretical aspects. We also show that other, slightly different, choices of the grid distributions already lead to a substantial degradation of the accuracy. The numerical results illustrate the effectiveness of the proposed higher order numerical strategy for nonlinear convection dominated singularly perturbed boundary value problems.