Asymptotically accurate error estimates of exponential convergence for the trapezoidal rule

In many applied problems, efficient calculation of quadratures with high accuracy is required. The examples are: calculation of special functions of mathematical physics, calculation of Fourier coefficients of a given function, Fourier and Laplace transformations, numerical solution of integral equations, solution of boundary value problems for partial differential equations in integral form, etc. For grid calculation of quadratures, the trapezoidal, the mean and the Simpson methods are usually used. Commonly, the error of these methods depends quadratically on the grid step, and a large number of steps are required to obtain good accuracy. However, there are some cases when the error of the trapezoidal method depends on the step value not quadratically, but exponentially. Such cases are integral of a periodic function over the full period and the integral over the entire real axis of a function that decreases rapidly enough at infinity. If the integrand has poles of the first order on the complex plane, then the Trefethen-Weidemann majorant accuracy estimates are valid for such quadratures. In the present paper, new error estimates of exponentially converging quadratures from periodic functions over the full period are constructed. The integrand function can have an arbitrary number of poles of an integer order on the complex plane. If the grid is sufficiently detailed, i.e., it resolves the profile of the integrand function, then the proposed estimates are not majorant, but asymptotically sharp. Extrapolating, i.e., excluding this error from the numerical quadrature, it is possible to calculate the integrals of these classes with the accuracy of rounding errors already on extremely coarse grids containing only 10 steps.

Discretization Error Estimation in Subsonic, Transonic and Supersonic Flows of an Inviscid Fluid Over a Bump

This paper presents a solution verification exercise for the simulation of subsonic, transonic and supersonic flows of an inviscid fluid over a circular arc (bump). Numerical simulations are performed with a pressure-based, single-phase compressible flow solver. Sets of geometrically similar grids covering a wide range of refinement ratios have been generated. The goal of these grids is twofold: obtain a reference solution from power series expansion fits applied to the finest grids; check the numerical uncertainties obtained from coarse grids that do not guarantee monotonic convergence of the quantities of interest. The results show that even with very fine grids it is not straightforward to define a reference solution from power series expansions. The level of discretization errors required to obtain reliable reference solutions implies iterative errors reduced to machine accuracy, which may be extremely time consuming even in two-dimensional inviscid flows. Quantitative assessment of the estimated uncertainties for coarse grids depends on the selected reference solution.

A Time-Space Multigrid Method for Efficient Solution of the Harmonic Balance Equation System

In this paper, an efficient time-space multigrid (TS-MG) method for solving a harmonic balance (HB) equation system is proposed. The principle of the time-space multigrid method is to coarsen grids in both space and time dimensions simultaneously when coarse grids are formed. The inclusion of time in the time-space multigrid is to address the instability issue or diminished convergence speedup of the spatial multigrid (S-MG) due to larger grid reduced frequencies on coarse grids. With the proposed method, the unsteady governing equation will be solved on all grid levels. Comparing to the finest grid, fewer harmonics and thus fewer equations will be solved consequently on coarse grids. Discrete Fourier transform (DFT) and inverse discrete Fourier transform (IDFT) are used to achieve solution prolongation and restriction between different time grid levels. Results from the proposed method are compared with those obtained from the traditional spatial multigrid and time domain methods. It is found that the TS-MG method can increase solution stability, reduce analysis time cost required for convergence, save memory usage and has no adverse effect on solution accuracy.

A global geographic grid system for visualizing bathymetry

A global geographic grid system (Global GGS) is here introduced to support the display of gridded bathymetric data at whatever resolution is available in a visually seamless manner. The Global GGS combines a quadtree metagrid hierarchy with a system of compatible data grids. Metagrid nodes define the boundaries of data grids. Data grids are regular grids of depth values, coarse grids are used to represent sparse data and finer grids are used to represent high-resolution data. Both metagrids and data grids are defined in geographic coordinates to allow broad compatibility with the widest range of geospatial software packages. An important goal of the Global GGS is to support the meshing of adjacent tiles with different resolutions so as to create a seamless surface. This is accomplished by ensuring that abutting data grids either match exactly with respect to their grid-cell size or only differ by powers of 2. The oversampling of geographic data grids, which occurs towards the poles due to the convergence of meridians, is addressed by reducing the number of columns (longitude sampling) by powers of 2 at appropriate lines of latitude. In addition to the specification of the Global GGS, this paper describes a proof-of-concept implementation and some possible variants.

Global-to-Cartesian 3-D EM modeling using a nested IE approach with application to long-period responses from island geomagnetic observatories

<p>There is a significant interest in constraining the mantle conductivity beneath oceans. One of the main sources of data that can be used to reveal the conductivity distribution in the oceanic mantle are time-varying magnetic fields measured at island geomagnetic observatories. From these data local electromagnetic (EM) responses are estimated and then inverted in terms of conductivity. The challenge here is that island responses are strongly distorted by the ocean induction effect (OIE) originating from the lateral conductivity contrasts between the conductive ocean and resistive land. OIE is generally modeled by global simulations using relatively coarse grids (down to 0.25 degree resolution) to represent the bathymetry. Insufficiently accurate accounting for the OIE may lead to the wrong interpretation of the observed responses. We study whether the small-scale bathymetry features influence the island responses. To address this question we developed a global-to-Cartesian 3-D EM modeling framework based on a nested integral equation approach, which allows to efficiently account for the effects of high-resolution bathymetry. Two geomagnetic observatories, located in Indian (Cocos Island) and Pacific (Oahu Island) Oceans, are chosen to study the OIE in long-period responses. Numerical tests demonstrate that accounting of the very local bathymetry (down to 1 km resolution) dramatically change modeling results. Remarkably, the anomalous behavior of the imaginary parts of the responses at Cocos Island, namely, the change of sign at short periods, is reproduced by using highly detailed bathymetry.</p>

A Global Geographic Grid System for Visualizing Bathymetry

A Global Geographic Grid System (Global GGS) is here introduced to support the display of gridded bathymetric data at whatever resolution is available in a visually seamless manner. The Global GGS combines a quad-tree metagrid hierarchy with a system of compatible data grids. Metagrid nodes define the boundaries of data grids. Data grids are regular grids of depth values, coarse grids are used to represent sparse data and finer grids are used to represent high resolution data. Both metagrids and data grids are defined in geographic coordinates to allow broad compatibility with the widest range of geospatial software packages. An important goal of the Global GGS is to support the meshing of adjacent tiles with different resolutions so as to create a seamless surface. This is accomplished by ensuring that abutting data grids either match exactly with respect to their grid-cell size or only differ by powers of two. The oversampling of geographic data grids, which occurs towards the poles due to the convergence of meridians, is addressed by reducing the number of columns (longitude sampling) by powers of two at appropriate lines of latitude. In addition to the specification of the Global GGS. This paper describes a proof-of-concept implementation and some possible variants.

A Novel Edge-Based Green Element Method for Simulating Fluid Flow in Unconventional Reservoirs with Discrete Fractures

Summary The boundary-element method (BEM) is widely used in modeling fluid flow in fractured reservoirs. However, the computation is extremely expensive when real heterogeneity and large numbers of fractures are modeled. This paper presents a novel edge-based Green element method (GEM) (eGEM) for this problem, and two significant modifications are made to the classical GEM. An edge-based discretization scheme is proposed to improve accuracy of the GEM. The eGEM technique is further enriched for simulating discrete fractures. The mathematical model is transformed into the Laplace domain, which makes it convenient to incorporate multiporosity models because the form of the boundary integral equation is the same. The matrix is meshed using Cartesian grids, and discrete fractures are handled flexibly by embedding into the matrix grids. In eGEM, the matrix/matrix flow is coupled at the common edge, so the unknown flux can be eliminated by using the edge-based scheme. In each matrix block, the matrix/fracture flow is modeled by treating the fracture elements as sources or sinks, as with BEM. The finite-difference method (FDM) is used to handle the fracture/fracture flow. In this paper, we tested the numerical accuracy and computational efficiency of the eGEM using several cases. First, the technique was shown to have higher accuracy than the classical corner-based GEM for transient problems in the petroleum industry. This shows the advantage of the edge-based discretization approach in handling the unknown flux of each solution point. The ability of the eGEM to handle discrete fractures was validated with the several models for transient-flow problems. The computation of the proposed eGEM is much less expensive than that for BEM in modeling the transient behavior of fractured media. Compared with the commercial numerical simulator in handling discrete fractures, the eGEM is shown to be less grid sensitive and to maintain a relatively high precision even with coarse grids near the discrete fractures. A detailed grid-sensitivity analysis was performed. The fracture grids are recommended to be refined to capture the early-time-flow behavior in pressure-transient analysis, especially in modeling low-conductivity fractures. For the first time, an efficient edge-based discretization scheme for GEM is presented, which handles the unknown flux of each solution point and at the same time uses eGEM to enrich simulation of discrete-fracture networks. This method serves as a new efficient approach for reservoir simulation and numerical well testing. Because of the high precision of eGEM with coarse grids, it would be efficient in larger field applications.