Non-uniform grid finite-difference seismic wave simulation using multiblock grids by adding positive and negative singularity pairs

This paper compares horizontal well performance simulated by a uniform coarse grid, a uniform fine grid, a nonuniform fine grid using the Peaceman's well model, and a non-uniform fine grid with explicit well modeling. An analytical solution is used to evaluate numerical solutions. We find that when a coarse grid is used to simulate a partially penetrating horizontal well, the fiow rate of the well is under-predicted, which means the well index calculated by Peaceman's method needs to be increased for the numerical simulator to predict the well productivity correctly. Peaceman's well model was developed for two-dimensional flow problems. Since these assumptions are violated for a partially penetrating horizontal well, Peaceman's formula does not accurately predict well performance. On the other hand, explicit modeling is a technique where the well blocks are of the same size as the well so that the well block pressure can be considered as the wellbore pressure. This eliminates the need for using a well index. However, our analysis shows that the non-uniform fine grid used for explicit modeling of wells, as reported in a recent paper, can lead to incorrect results. This is mainly because nonuniform grids can introduce large numerical errors into the computation of well block pressure which is made to be the same as the wellbore pressure, which is the explicit modeling technique. We demonstrate that the best way to obtain the correct well productivity is to use an appropriate analytical solution. When an analytical solution is not available, a uniform fine grid plus explicit well modeling can also yield correct results, but with large computational expense. We show that for such situations, use of non-uniform grids plus explicit modeling with uniform local grid refinement gives satisfactory results with reasonable computational costs. In this paper we also propose practical methods for calculating the correct well index.

The use of non-uniform grids in the MoM applied to electromagnetic scattering has been presented. After writing the classical MoM in an operator formulation, it was shown how a non-uniform grid could be built and how it can decrease the size of the algebraic system to be solved. Then, the utility of the transfinite interpolation was shown in order to satisfy the boundary conditions in the non-uniform grid. Finally, applications to electromagnetic scattering by biological tissues were shown. Although this method has been studied for 2D geometries, it could be developed in three-dimensional cases.

The finite difference (FD) method is popular in the computational fluid dynamics and widely used in various flow simulations. Most of the FD schemes are developed on the uniform Cartesian grids; however, the use of nonuniform or curvilinear grids is inevitable for adapting to the complex configurations and the coordinate transformation is usually adopted. Therefore the question that whether the characteristics of the numerical schemes evaluated on the uniform grids can be preserved on the nonuniform grids arises, which is seldom discussed. Based on the one-dimensional wave equation, this paper systematically studies the characteristics of the high-order FD schemes on nonuniform grids, including the order of accuracy, resolution characteristics and the numerical stability. Especially, the Fourier analysis involving the metrics is presented for the first time and the relation between the resolution of numerical schemes and the stretching ratio of grids is discussed. Analysis shows that for smooth varying grids, these characteristics can be generally preserved after the coordinate transformation. Numerical tests also validate our conclusions.

As is well known, one of the main challenges in dealing with the time‐fractional diffusion‐wave equation is that the solutions have weak regularity at the initial values. This necessitates the use of nonuniform grids for error analysis. However, the theoretical analysis of nonuniform grids is relatively complex and lacks efficient and stable numerical methods. The main research problem addressed in this article is the time‐fractional diffusion‐wave equation on nonuniform grids. We use the order reduction method and discrete complementary convolution kernel to discretize the Caputo derivative of the equation and obtain the L1‐type finite element method on nonuniform grids. The properties related to discrete complementary convolutional kernels are often used for algorithm convergence analysis, which simplifies the process of finite element theory analysis and is efficient and innovative. This article demonstrates the solvability, stability, and convergence of the L1‐type finite element schemes on nonuniform grids. Finally, the consistency between the proposed finite element scheme and the convergence order results obtained from theoretical analysis was verified through experiments.

Grid-optimized upwind dispersion-relation-preserving scheme on non-uniform Cartesian grids for computational aeroacoustics

The grid orientation effect is a long-standing problem plaguing reservoir simulators that employ finite difference schemes for solution. The nine-point finite difference scheme is a well known way of reducing this effect. The schemes available in the literature are applicable only to point distributed grid systems. However, most simulators used in practice employ block centered grid systems. In this report we present a derivation of a new nine-point scheme for the block centered grid systems. The derivation is based on physical consideration and hence is free from any problems such as asymmetry or negativity of interblock transmissibilities that arise in some published schemes. The scheme is applicable to nonuniform rectangular grids and to reservoirs with inhomogeneous permeability distributions; both isotropic and anisotropic cases are treated. The expressions resulting from the new scheme reduce to those published in the literature when a square uniform grid and homogeneous isotropic permeability are considered. The reduction in the grid orientation effect due to the new scheme is illustrated by a study of the steam-flood process in a five-spot pattern.

In this paper, we discuss the use of nonuniform grids in reduced basis design for developing low order nonlinear feedback controllers for hybrid distributed parameter systems. The reduced basis approach was presented in an earlier paper by Burns and King; therein, all approximations were based upon uniform grids. In this paper, we explore the effect on control design of using nonuniform grids in the fundamental step of approximating the functional controller gains. We illustrate the process using a weakly nonlinear distributed parameter system.

A compact streamfunction–velocity scheme on nonuniform grids for the 2D steady incompressible Navier–Stokes equations

Comparisons of compact and classical finite difference solutions of stiff problems on nonuniform grids

Objectives. Integral equations have long been used in mathematical physics to demonstrate existence and uniqueness theorems for solving boundary value problems for differential equations. However, despite integral equations have a number of advantages in comparison with corresponding boundary value problems where boundary conditions are present in the kernels of equations, they are rarely used for obtaining numerical solutions of problems due to the presence of equations with dense matrices that arise that when discretizing integral equations, as opposed to sparse matrices in the case of differential equations. Recently, due to the development of computer technology and methods of computational mathematics, integral equations have been used for the numerical solution of specific problems. In the present work, two methods for numerical solution of two-dimensional and three-dimensional integral equations are proposed for describing several significant classes of problems in mathematical physics.Methods. The method of collocation on non-uniform and uniform grids is used to discretize integral equations. To obtain a numerical solution of the resulting systems of linear algebraic equations (SLAEs), iterative methods are used. In the case of a uniform grid, an efficient method for multiplying the SLAE matrix by vector is created.Results. Corresponding SLAEs describing the considered classes of problems are set up. Efficient solution algorithms using fast Fourier transforms are proposed for solving systems of equations obtained using a uniform grid.Conclusions. While SLAEs using a non-uniform grid can be used to describe complex domain configurations, there are significant constraints on the dimensionality of described systems. When using a uniform grid, the dimensionality of SLAEs can be several orders of magnitude higher; however, in this case, it may be difficult to describe the complex configuration of the domain. Selection of the particular method depends on the specific problem and available computational resources. Thus, SLAEs on a non-uniform grid may be preferable for many two-dimensional problems, while systems on a uniform grid may be preferable for three-dimensional problems.

Stability of Saul'yev's methods for heat conduction with nonuniform grids is investigated. Though these methods are known to be unconditionally stable with uniform grids, the author shows that their stability with nonuniform grids depends on time step Δ τ, space intervals Δ X, and ratios of neighboring space intervals. The author also shows that physical reality is broken when Patankar's positive-coefficients rule is not satisfied, even if the methods are applied to uniform grids. This article presents a stability criterion for Saul'yev's methods with both nonuniform and uniform grids.

Full-wave analysis of the microstrip structures is performed by using the compact two-dimensional (2-D) finite-difference frequency-domain (FDFD) method with nonuniform grids and perfectly matched layer (PML). The use of nonuniform grids can significantly reduce the computational matrix size. Less memory and CPU time are required as comparing with the original compact 2-D FDFD method. For the analysis of the microstrip structures with an absorbing boundary condition, the compact 2-D FDFD method with PML is presented. The performances of different PML thickness are studied. Numerical examples are presented to demonstrate the accuracy and efficiency of this method.

Accurate and efficient numerical wave propagation is important in many areas of study such as computational aero-acoustics (CAA). While dissipation and dispersion errors influence the accuracy of a method, efficiency can be assessed by convergence rates and effective adaptability to different mesh structures. Finite difference and finite element methods are commonly used numerical schemes in CAA. Finite difference methods have the advantages of ease of use as well as high order convergence, but often require a uniform grid, and stable boundary closure can be non-trivial. Finite element methods adapt well to different mesh structures but can become difficult to implement as the order of approximation increases. In this research we formulate a numerical method that has high-order convergence, with strong accuracy for numerical wave numbers, and is adaptive to non-uniform grids. Such a method is developed based on the Discontinuous Galerkin Method (DGM) applied to the hyperbolic equation. Finite difference type schemes applicable to non-uniform grids are proposed. The schemes will be referred to as DGM-FD schemes. These schemes inherit, naturally, some features of the DGM, such as high-order approximations, applicability to non-uniform grids and super-accuracy for wave propagations. Two grid structures are studied. In the first structure, a regular, but non-uniform, finite difference type grid is assumed. In the second structure, some grid points are double-valued and the derivative scheme has a shortened stencil. Fourth-order upwind and third order central schemes are presented as examples of the first grid structure. Fifth-order upwind schemes are derived for the second structure. For non-linear equations, flux finite difference formula are given where no explicit upwind and downwind split of the flux is needed. This is in contrast to existing upwind finite difference schemes in the literature. Stability of the schemes with boundary closures and the super-accuracy for wave propagation problems are investigated and validated. The new schemes are demonstrated by numerical examples including the linearized acoustic waves, the solution of non-linear Burger's equation and the flat-plate boundary layer problem.

In the present work, we introduce a finite difference scheme on an nonuniform grid. The truncation errors introduced by the use of this difference scheme is presented. It is shown that the numerical solution in the physical domain on nonuniform grids has some advantages. Finally, we solve some boundary value problems using the introduced scheme and compare the obtained results with that obtained on an uniform grid.

An accurate and robust finite volume scheme based on the spline interpolation for solving the Euler and Navier–Stokes equations on non-uniform curvilinear grids