Tailored Finite Point Method Research Articles

Streamlined numerical solutions of burgers' equations bridging the tailored finite point method and the Cole Hopf transformation

This paper explores the possibility of solving the one-dimensional Burgers' equation with a tailored finite point method (TFPM) based on an explicit stencil along with utilizing the Cole Hopf transformation. The proposed TFPM procedure operates on an explicit four-point stencil where the nodal solution at the advanced temporal level is written as a linear combination of the nodal solutions at the preceding temporal level. In order to bring the essence of the local exact solutions into the numerical approximations the scalar coefficients involved in the linear sum are determined by the application of the fundamental solutions into the stencil. The efficiency of the method is demonstrated through the comparisons of TFPM solutions for various examples with the exact solutions and solutions from well-established methodologies in the literature.

An implicit tailored finite point method for the Burgers’ equation: leveraging the Cole-Hopf transformation

Any expedition in designing numerical methods besides aiming at accuracy, also equally steers for simplicity and ease in implementation. This paper brings in one such algorithm the tailored finite point method (TFPM) in tandem with the Cole Hopf transformation. At the initiation, the non-linear Burgers’ equation is transformed into a linear heat equation to which TFPM is applied. The proffered TFPM functions on an explicit stencil on the left boundary of the domain and on an implicit stair stencil throughout the rest of the domain. On these stencils, the nodal solutions at the advanced temporal level are written as a linear combination of the solutions at the remaining nodes within the stencil. The scalars involved in the linear combination are identified by the application of fundamental solutions into the stencil resultantly infusing the essential nature of the local exact solutions into the approximations. The foundation of such a linear combination avoids the need for complex computations involving matrix multiplication and inversion. The numerical accuracy of the method is established through comparisons of TFPM solutions of classical examples with the exact solutions and solutions from other contemporary methodologies. The theoretical correctness of the method is established through analyses of consistency, stability, and convergence. Furthermore, the method exhibits the potential for extension to higher dimensions and other complex modalities.

Image Super-Resolution Reconstruction by Huber Regularization and Tailored Finite Point Method

Image Super-Resolution Reconstruction by Huber Regularization and Tailored Finite Point Method

Two linear finite difference schemes based on exponential basis for two-dimensional time fractional Burgers equation

In present paper, we propose tailored finite point method (TFPM) based on exponential basis for solving two-dimensional time fractional Burgers equation. We use the L1 and L1-2 discretization formulas for the Caputo fractional derivative in time direction and use the TFPM to approximate the spatial derivatives. This method with the help of exponential functions fit the properties of the local solution in time and space simultaneously which act as basis functions in the frame of TFPM. We focus on constructing two linear implicit finite difference schemes, i.e. a five-point node centered scheme and a four-point cell-centered scheme based on exponential functions for deriving the numerical solution of given problem. The stability and convergence of fully discrete schemes based on five-point node centered are discussed. At last, we use some numerical examples to show the accuracy and efficiency of the presented schemes.

Uniformly convergent scheme for steady MHD duct flow problems with high Hartmann numbers on structured and unstructured grids

The steady incompressible Magnetohydrodynamic (MHD) duct flow problems with a transverse magnetic field at high Hartmann numbers (Ha) is a coupled system of convection-dominated diffusion equations and the solutions contain sharp boundary layers. We develop a five-point scheme and an edge-centered scheme on structured and unstructured grids with tailored finite point method (TFPM) for solving the coupled system of convection–diffusion equations. The five-point scheme is an upwind difference scheme, where the coefficients of the difference operator are tailored to some particular properties of the convection–diffusion equation with constant coefficients on the local cell. The edge-centered scheme is constructed by the continuity of the normal fluxes for each edge. The normal flux is expressed in terms of the edge-centered unknowns defined at the local cell and the unknowns at the adjacent cells are not needed. Uniform second order convergence can be obtained by the edge-centered scheme on structured and unstructured grids over a wide range of Ha 10 to 106. Both of the two schemes can correctly reproduce the fast change of the solutions which contain sharp boundary layers at high Ha even without refining the mesh. Numerical examples are provided to show the accuracy and of the TFPM.

An edge-centered scheme for anisotropic diffusion problems with discontinuities on distorted quadrilateral meshes

We propose a Tailored Finite Point Method for the solution of transient anisotropic diffusion problems on distorted quadrilateral meshes. We establish an edge-centered scheme based on the flux continuity. The main feature of the scheme lies in the fact that the approximate solutions as well as the gradients are determined by using a linear combination of special solutions. These special solutions are required to satisfy the homogeneous form of approximated equations on the local cell. This brings two important assets to numerical solution of the problems. First, the scheme efficiently tailors itself fit into the local properties of the solution, so that it exhibits robustness to mesh distortion and can correctly reproduce the details of the interface layers even without refining the mesh. Second, the flux can be approximated in terms of the edge-centered unknowns defined at the local cell and the unknowns at the adjacent cells are not needed. The local stencil makes writing code for the continuity condition of the flux on each edge easier. Numerical experiments are carried out to validate the performance of the scheme with or without discontinuities on various types of distorted quadrilateral meshes.

Tailored Finite Point Method for Diffusion Equations with Interfaces on Distorted Meshes

Tailored Finite Point Method for Diffusion Equations with Interfaces on Distorted Meshes

A Cartesian grid based tailored finite point method for reaction-diffusion equation on complex domains

This paper presents a Cartesian grid based tailored finite point method (TFPM) for singularly perturbed reaction-diffusion equation on complex domains. The method is incorporated with the kernel-free boundary integral algorithm, where the semi-discrete boundary value problems after time integration are reformulated into corresponding Fredholm boundary integral equations (BIEs) of the second kind, however with no algorithmic dependence on the exact analytical expression for the kernels of integrals. The BIEs are iteratively solved by the GMRES method while integral evaluation during each iteration resorts to solving an equivalent interface problem, which in practice is achieved by a series of manipulations in the framework of TFPM including discretization, correction, solution, and interpolation. The proposed method has second-order accuracy for the reaction-diffusion equation as demonstrated by the numerical examples.

A tailored finite point method for subdiffusion equation with anisotropic and discontinuous diffusivity

In this paper, we first propose a tailored finite point method (TFPM) for solving time fractional subdiffusion problems with anisotropic and discontinuous diffusivity. This numerical scheme can perfectly capture the rapid transition of the solutions which contain sharp interface layers even with coarse meshes. Second, the accuracy and stability of the proposed scheme are strictly analyzed. Finally, some numerical examples are provided to show the accuracy and reliability of this new scheme.

Novel finite point approach for solving time-fractional convection-dominated diffusion equations

In this paper, a stabilized numerical method with high accuracy is proposed to solve time-fractional singularly perturbed convection-diffusion equation with variable coefficients. The tailored finite point method (TFPM) is adopted to discrete equation in the spatial direction, while the time direction is discreted by the G-L approximation and the L1 approximation. It can effectively eliminate non-physical oscillation or excessive numerical dispersion caused by convection dominant. The stability of the scheme is verified by theoretical analysis. Finally, one-dimensional and two-dimensional numerical examples are presented to verify the efficiency of the method.

Tailored finite point method for the approximation of diffusion operators with non-symmetric diffusion tensor

We present a tailored finite point method (TFPM) for anisotropic diffusion equations with a non-symmetric diffusion tensor on Cartesian grids. The fluxes on each edge are discretized by using a linear combination of the local basis functions, which come from the exact solution of the diffusion equation with constant coefficients on the local cell. In this way, the scheme is fully consistent and the flux is naturally continuous across the interfaces between the subdomains with a non-symmetric diffusion tensor. Additionally, it is convenient to handle the Neumann boundary condition or a variant of Neumann boundary condition. Numerical results obtained from solving different anisotropic diffusion problems including problems with sharp discontinuity near the interface and boundary, show that this approach is efficient. Second order convergence rate can be obtained with the numerical examples. The new scheme is also tested on a time-dependent problem modeling the Hall effect in the resistive magnetohydrodynamics, the results further show the robustness of this new method.

Tailored finite point method for time fractional convection dominated diffusion problems with boundary layers

We propose a tailored finite point method (TFPM) for solving time fractional convection dominated diffusion equations in this paper. The main idea of TFPM is to firstly approximate the diffusion, convection coefficient near each grid by a constant, and then determine the weights of the finite difference scheme by using the exact solution of the convection diffusion equation with constant coefficients. This adaptation perfectly captures the rapid transition of the solutions which contain sharp boundary layers even with coarse meshes. The accuracy and stability of the scheme are rigorously analyzed. Numerical examples are shown to verify the accuracy and reliability of the proposed scheme.

A new tailored finite point method for strongly anisotropic diffusion equation on misaligned grids

This paper presents a new tailored finite point method (TFPM) for the strongly anisotropic diffusion equation on Cartesian coordinates. The novelty is that the scheme is constructed by using the interface conditions for each cell. Several numerical experiments are presented to show the performance of this new scheme. Numerically, the method can not only achieve good accuracy, but also sharply capture internal layers.

Two uniform tailored finite point method for strongly anisotropic and discontinuous diffusivity

The paper presents two Tailored Finite Point method (TFPM) for the diffusion equations, which is valid for the strongly anisotropic tensor diffusivity and interface layers. The first scheme uses the value as well as their derivatives at the grid points to construct the five point scheme for the heterogeneous rotating anisotropy. The second scheme is based on the interface conditions to construct the four point scheme for each cell, which gives rise to desirable internal layers and discontinuous diffusivity conditions. Numerically, both methods can achieve uniform convergence, even when there exhibit interface layers. Numerical experiments are presented to show the performance of the proposed two different schemes.

An Efficient Tailored Finite Point Method for Rician Denoising and Deblurring

An Efficient Tailored Finite Point Method for Rician Denoising and Deblurring

Uniform Convergent Tailored Finite Point Method for Advection–Diffusion Equation with Discontinuous, Anisotropic and Vanishing Diffusivity

We propose two tailored finite point methods for the advection---diffusion equation with anisotropic tensor diffusivity. The diffusion coefficient can be very small in one direction in some part of the domain and be discontinuous at the interfaces. When flows advect from the vanishing-diffusivity region towards the non-vanishing diffusivity region, standard numerical schemes tend to cause spurious oscillations or negative values. Our proposed schemes have uniform convergence in the vanishing diffusivity limit, even when the solution exhibits interface and boundary layers. When the diffusivity is along the coordinates, the positivity and maximum principle can be proved. We use the value as well as their derivatives at the grid points to construct the scheme for nonaligned case, which makes it can achieve good accuracy and convergence for the derivatives as well, even when there exhibit boundary or interface layers. Numerical experiments are presented to show the performance of the proposed scheme.

An Anisotropic Convection-Diffusion Model Using Tailored Finite Point Method for Image Denoising and Compression

AbstractIn this paper we consider an anisotropic convection-diffusion (ACD) filter for image denoising and compression simultaneously. The ACD filter is discretized by a tailored finite point method (TFPM), which can tailor some particular properties of the image in an irregular grid structure. A quadtree structure is implemented for the storage in multi-levels for the compression. We compare the performance of the proposed scheme with several well-known filters. The numerical results show that the proposed method is effective for removing a mixture of white Gaussian and salt-and-pepper noises.

Tailored Finite Point Method for Parabolic Problems

Abstract In this paper, we propose a class of new tailored finite point methods (TFPM) for the numerical solution of parabolic equations. Our finite point method has been tailored based on the local exponential basis functions. By the idea of our TFPM, we can recover all the traditional finite difference schemes. We can also construct some new TFPM schemes with better stability condition and accuracy. Furthermore, combining with the Shishkin mesh technique, we construct the uniformly convergent TFPM scheme for the convection-dominant convection-diffusion problem. Our numerical examples show the efficiency and reliability of TFPM.

Tailored finite point method for solving one-dimensional Burgers' equation

ABSTRACTWe propose a tailored finite point method (TFPM) for solving a quasilinear time-dependent Burgers' equation with a small coefficient of viscosity. The selected basis functions for the TFPM automatically fit the properties of the local solution in time and space simultaneously. The stability and error analysis for the TFPM are given. We also demonstrate the efficiency of the proposed scheme on relatively coarse meshes. The numerical results indicate that the TFPM achieves high accuracy and effectively captures the shock solutions.

An Equation Decomposition Based Tailored Finite Point Method for Linearized Incompressible Flow in Two-Dimensional Space

Abstract In this paper, we propose a tailored finite point method for linearized incompressible flow (Oseen equations) in two dimensions based on the equation decomposition technique. Unlike the usual vorticity-stream function formulation, the velocities are decomposed to irrotational and rotational parts. We only need to solve a system of two elliptic equations which are decoupled in the interior domain. They are only coupled in boundary conditions. When the domain is unbounded, we use the artificial boundary method to reduce the original problem to a problem on a bounded computational domain. Our finite point method has been tailored to some particular properties of the problem. Therefore, our scheme satisfies the discrete maximum principle in the interior domain automatically. We also give some remarks on more generally linearized incompressible flow, and it can be considered as the first step to solve the incompressible Navier–Stokes problem. Finally, several numerical examples show the efficiency and feasibility of our method whatever the Reynolds number is small or large.