A new finite-difference solution-adaptive method

1992 ◽  
Vol 341 (1662) ◽  
pp. 373-410 ◽  
Keyword(s):  
Finite Difference ◽  
Matrix Equation ◽  
Rapid Change ◽  
Penalty Functions ◽  
Hyperbolic Partial Differential Equations ◽  
Test Problems ◽  
Rarefaction Waves ◽  
Time Step ◽  
Error Measure ◽  
Flow Variables

A new moving finite difference (MFD) method has been developed for solving hyperbolic partial differential equations and is compared with the moving finite element (MFD) method of K. Miller and R. N. Miller. These methods involve the adaptive movement of nodes so as to reduce the number of nodes needed to solve a problem; they are applicable to the solution of non-stationary flow problems that contain moving regions of rapid change in the flow variables, surrounded by regions of relatively smooth variation. Both methods solve simultaneously for the flow variables and the node locations at each time-step, and they move the nodes so as to minimize an ‘error’ measure that contains a function of the time derivatives of the solution. This error measure is manipulated to obtain a matrix equation for node velocities. Both methods make use of penalty functions to prevent node crossing. The penalty functions result in extra terms in the matrix equation that promote node repulsion by becoming large when node separation becomes small. Extensive work applying the MFE and MFD methods to one-dimensional gasdynamic problems has been conducted to evaluate their performance. The test problems include Burgers’ equation, ideal viscid planar flow within a shock-tube, propagation of shock and rarefaction waves through area changes in ducts, and viscous transition through a contact surface and a shock.

scholarly journals THE NUMERICAL TREATMENT OF NONLINEAR FRACTAL–FRACTIONAL 2D EMDEN–FOWLER EQUATION UTILIZING 2D CHELYSHKOV POLYNOMIALS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040042
Author(s):  
M. HOSSEININIA ◽  
M. H. HEYDARI ◽  
Z. AVAZZADEH
Keyword(s):  
Approximate Solution ◽  
Finite Difference ◽  
Test Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Time Step ◽  
Discrete Method ◽  
Difference Formula ◽  
Fowler Equation ◽  
Chelyshkov Polynomials

This paper develops an effective semi-discrete method based on the 2D Chelyshkov polynomials (CPs) to provide an approximate solution of the fractal–fractional nonlinear Emden–Fowler equation. In this model, the fractal–fractional derivative in the concept of Atangana–Riemann–Liouville is considered. The proposed algorithm first discretizes the fractal–fractional differentiation by using the finite difference formula in the time direction. Then, it simplifies the original equation to the recurrent equations by expanding the unknown solution in terms of the 2D CPs and using the [Formula: see text]-weighted finite difference scheme. The differentiation operational matrices and the collocation method play an important role to obtaining a linear system of algebraic equations. Last, solving the obtained system provides an approximate solution in each time step. The validity of the formulated method is investigated through a sufficient number of test problems.

scholarly journals A Method for Imposing Surface Stress and Heat Flux Conditions in Finite-Difference Models with Steep Terrain

2007 ◽  
Vol 135 (3) ◽  
pp. 906-917 ◽  
Author(s):  
C. C. Epifanio
Keyword(s):  
Finite Difference ◽  
Linear Systems ◽  
Surface Stress ◽  
Test Problems ◽  
Time Step ◽  
Terrain Following ◽  
Thermally Driven ◽  
Stress Boundary Condition ◽  
Steep Terrain ◽  
Stress Boundary

Abstract A numerical implementation of the surface stress boundary condition is presented for finite-difference models in which the terrain slope and curvature cannot necessarily be considered small. The method involves reducing the discretized stress condition in terrain-following coordinates to a pair of coupled linear systems for the two horizontal velocity components at the boundary. The linear systems are then solved iteratively at each model time step to provide the unique boundary values of velocity consistent with the specified values of the stress. Similar methods are used to prescribe the normal flux of heat across the boundary. A related method for imposing stress conditions in two-dimensional vorticity–streamfunction models is also discussed. The effectiveness of the boundary conditions is demonstrated through a series of test problems involving topographic wake flows and thermally driven flows on steep slopes. It is shown that the use of the conventional flat-boundary approximation can lead to substantial errors when the resolved topography is sufficiently steep.

scholarly journals Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 923 ◽  
Author(s):  
Abdul Ghafoor ◽  
Sirajul Haq ◽  
Manzoor Hussain ◽  
Poom Kumam ◽  
Muhammad Asif Jan
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Quadrature Rule ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Haar Wavelets ◽  
Test Problems ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

Advanced Weathervaning in Ice

2011 ◽  
Author(s):  
William Hidding ◽  
Guillaume Bonnaffoux ◽  
Mamoun Naciri
Keyword(s):  
Rapid Change ◽  
Ice Sheet ◽  
Model Tests ◽  
The Arctic ◽  
Mooring System ◽  
Sudden Increase ◽  
Time Step ◽  
Ice Loads ◽  
Level Ice ◽  
Drift Direction

The reported presence of one third of remaining fossil reserves in the Arctic has sparked a lot of interest from energy companies. This has raised the necessity of developing specific engineering tools to design safely and accurately arctic-compliant offshore structures. The mooring system design of a turret-moored vessel in ice-infested waters is a clear example of such a key engineering tool. In the arctic region, a turret-moored vessel shall be designed to face many ice features: level ice, ice ridges or even icebergs. Regarding specifically level ice, a turret-moored vessel will tend to align her heading (to weather vane) with the ice sheet drift direction in order to decrease the mooring loads applied by this ice sheet. For a vessel already embedded in an ice sheet, a rapid change in the ice drift direction will suddenly increase the ice loads before the weathervaning occurs. This sudden increase in mooring loads may be a governing event for the turret-mooring system and should therefore be understood and simulated properly to ensure a safe design. The paper presents ADWICE (Advanced Weathervaning in ICE), an engineering tool dedicated to the calculation of the weathervaning of ship-shaped vessels in level ice. In ADWICE, the ice load formulation relies on the Croasdale model. Ice loads are calculated and applied to the vessel quasi-statically at each time step. The software also updates the hull waterline contour at each time step in order to calculate precisely the locations of contact between the hull and the ice sheet. Model tests of a turret-moored vessel have been performed in an ice basin. Validation of the simulated response is performed by comparison with model tests results in terms of weathervaning time, maximum mooring loads, and vessel motions.

Upwind finite‐difference calculation of traveltimes

Geophysics ◽  
1991 ◽  
Vol 56 (6) ◽  
pp. 812-821 ◽  
Author(s):  
J. van Trier ◽  
W. W. Symes
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Conservation Law ◽  
Finite Difference Scheme ◽  
Eikonal Equation ◽  
Upwind Schemes ◽  
Similar Scheme ◽  
Difference Calculation ◽  
Flow Variables ◽  
Upwind Finite Difference

Seismic traveltimes can be computed efficiently on a regular grid by an upwind finite‐difference method. The method solves a conservation law that describes changes in the gradient components of the traveltime field. The traveltime field itself is easily obtained from the solution of the conservation law by numerical integration. The conservation law derives from the eikonal equation, and its solution depicts the first‐arrival‐time field. The upwind finite‐difference scheme can be implemented in fully vectorized form, in contrast to a similar scheme proposed recently by Vidale. The resulting traveltime field is useful both in Kirchhoff migration and modeling and in seismic tomography. Many reliable methods exist for the numerical solution of conservation laws, which appear in fluid mechanics as statements of the conservation of mass, momentum, etc. A first‐order upwind finite‐difference scheme proves accurate enough for seismic applications. Upwind schemes are stable because they mimic the behavior of fluid flow by using only information taken from upstream in the fluid. Other common difference schemes are unstable, or overly dissipative, at shocks (discontinuities in flow variables), which are time gradient discontinuities in our approach to solving the eikonal equation.

Efficient Two-dimensional Acoustic Wave Finite-Difference Numerical Simulation in Strongly Heterogeneous Media Using the Adaptive Mesh Refinement (AMR) Technique

Geophysics ◽  
2021 ◽  
pp. 1-76
Author(s):  
Chunli Zhang ◽  
Wei Zhang
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Adaptive Mesh Refinement ◽  
Heterogeneous Media ◽  
Mesh Refinement ◽  
Adaptive Mesh ◽  
Grid Spacing ◽  
Time Step ◽  
Velocity Models ◽  
The Stability

The finite-difference method (FDM) is one of the most popular numerical methods to simulate seismic wave propagation in complex velocity models. If a uniform grid is applied in the FDM for heterogeneous models, the grid spacing is determined by the global minimum velocity to suppress dispersion and dissipation errors in the numerical scheme, resulting in spatial oversampling in higher-velocity zones. Then, the small grid spacing dictates a small time step due to the stability condition of explicit numerical schemes. The spatial oversampling and reduced time step will cause unnecessarily inefficient use of memory and computational resources in simulations for strongly heterogeneous media. To overcome this problem, we propose to use the adaptive mesh refinement (AMR) technique in the FDM to flexibly adjust the grid spacing following velocity variations. AMR is rarely utilized in acoustic wave simulations with the FDM due to the increased complexity of implementation, including its data management, grid generation and computational load balancing on high-performance computing platforms. We implement AMR for 2D acoustic wave simulation in strongly heterogeneous media based on the patch approach with the FDM. The AMR grid can be automatically generated for given velocity models. To simplify the implementation, we employ a well-developed AMR framework, AMReX, to carry out the complex grid management. Numerical tests demonstrate the stability, accuracy level and efficiency of the AMR scheme. The computation time is approximately proportional to the number of grid points, and the overhead due to the wavefield exchange and data structure is small.

Tests of the Stability and Time-Step Sensitivity of Semi-Implicit Reservoir Stimulation Techniques

1972 ◽  
Vol 12 (03) ◽  
pp. 253-266 ◽  
Author(s):  
James S. Nolen ◽  
D.W. Berry
Keyword(s):  
Finite Difference ◽  
Difference Equations ◽  
Implicit Method ◽  
Significant Advance ◽  
Relative Permeabilities ◽  
Time Step ◽  
Finite Difference Equations ◽  
Nonlinear Form ◽  
Implicit Finite Difference ◽  
Excellent Stability

Abstract A reservoir simulation technique that employs semi-implicit approximations to relative permeabilities exhibits excellent stability and permeabilities exhibits excellent stability and convergence characteristics when applied to water- or gas-coning problems. Recent workers in this area have made a simplifying assumption in order to linearize the flow terms of the semi-implicit finite-difference equations. This paper describes a method of solving efficiently paper describes a method of solving efficiently the nonlinear form of the equations and demonstrates that time-step sensitivity is reduced by iterating on the nonlinear terms. In addition, it addresses the problem of allocating a well's production among multiple grid blocks. Example problems include both water-coning and gas-percolation applications. Introduction Multiphase reservoir simulators traditionally have employed finite-difference approximations in which relative permeabilities are evaluated explicitly at the beginning of each time step. Simulators of this type are capable of handling many reservoir studies in a perfectly satisfactory fashion, but they are incapable of solving economically problems characterized by high flow velocities. Included in this category are the studies of such phenomena as well coning and gas percolation. The difficulty in such problems is a stability limitation imposed by the use of explicit relative permeabilities. In an attempt to overcome this permeabilities. In an attempt to overcome this limitation, Blair and Weinaug developed a simulator that employed implicitly evaluated relative permeabilities. The increased stability of their permeabilities. The increased stability of their equations allowed the use of time steps much larger than previously possible, but this was counteracted by an increase in the computational work per time step and an increased difficulty in the iterative solution of the difference equations. While the net result was a significant advance in the solution of coning problems, improvements still were needed to increase the dependability and decrease the cost of obtaining solutions for such problems. More recently, two papers were published describing a method that employs semi-implicit relative permeabilities. This method is greatly superior to the fully implicit method, both in computational effort and maximum time-step size. In developing this method, the previous workers made a simplifying assumption to obtain linear finite-difference equations. We have developed a reservoir simulator based on the nonlinear form of the semi-implicit finite-difference equations. This paper describes the techniques used in the simulator and presents the results of some tests conducted with it. These include time-step sensitivity studies and tests of alternate production allocation methods. Some of these tests compare the nonlinear form of the semi-implicit method with the linear form. SPEJ P. 253

Practical numerical considerations for the Alekseev–Mikhailenko method

1990 ◽  
Vol 27 (8) ◽  
pp. 1023-1030 ◽  
Author(s):  
P. F. Daley ◽  
F. Hron
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Integral Transforms ◽  
Hankel Transform ◽  
Radial Symmetry ◽  
Hyperbolic Partial Differential Equations ◽  
Finite Integral ◽  
Finite Hankel Transform ◽  
Difference Methods ◽  
New Generation

Programs that utilize the Alekseev–Mikhailenko method are becoming viable seismic interpretation aids because of the availability of a new generation of supercomputers. This method is highly numerically accurate, employing a combination of finite integral transforms and finite difference methods, for the solution of hyperbolic partial differential equations, to yield the total seismic wave field.In this paper two questions of a numerical nature are addressed. For coupled P–Sv wave propagation with radial symmetry, Hankel transforms of order 0 and 1 are required to cast the problem in a form suitable for solution by finite difference methods. The inverse series summations would normally require that the two sets of roots of the transcendental equations be employed, corresponding to the zeroes of the Bessel functions of order 0 and 1. This matter is clarified, and it is shown that both inverse series summations may be performed by considering only one set of roots.The second topic involves providing practical means of determining the lower and upper bounds of a truncated series that suitably approximates the infinite inverse series summation of the finite Hankel transform. It is shown that the number of terms in the truncated series generally decreases with increasing duration of the source pulse and that the truncated series may be further reduced if near-vertical-incidence seismic traces are avoided.

A high order finite-difference scheme for time-domain modelling of time-varying seismoelectric waves

Geophysics ◽  
2021 ◽  
pp. 1-49
Author(s):  
Yanju Ji ◽  
Li Han ◽  
Xingguo Huang ◽  
Xuejiao Zhao ◽  
Kristian Jensen ◽  
...  
Keyword(s):  
Porous Media ◽  
Finite Difference ◽  
Time Domain ◽  
Calculation Method ◽  
High Order ◽  
Difference Approximation ◽  
Time Varying ◽  
Time Step ◽  
Static Approximation ◽  
High Order Finite Difference

Simulation of the seismoelectric effect serves as a useful tool to capture the observed seismoelectric conversion phenomenon in porous media, thus offering promising potential in underground exploration activities to detect pore fluids such as water, oil and gas. The static electromagnetic (EM) approximation is among the most widely used methods for numerical simulation of the seismoelectric responses. However, the static approximation ignores the accompanying electric field generated by the shear wave, resulting in considerable errors when compared to analytical results, particularly under high salinity conditions. To mitigate this problem, we propose a spatial high-order finite-difference time-domain (FDTD) method based on Maxwell's full equations of time-varying EM fields to simulate the seismoelectric response in 2D mode. To improve the computational efficiency influenced by the velocity differences between seismic and electromagnetic waves, different time steps are set according to the stability conditions, and the seismic feedback values of EM time nodes are obtained by linear approximation within the seismic unit time step. To improve the simulation accuracy of the seismoelectric response with the time-varying EM calculation method, finite-difference coefficients are obtained by solving the spatial high-order difference approximation based on Taylor expansion. The proposed method yields consistent simulation results compared to those obtained from the analytical method under different salinity conditions, thus indicating its validity for simulating seismoelectric responses in porous media. We further apply our method to both layered and anomalous body models and extend our algorithm to 3D. Results show that the time-varying EM calculation method could effectively capture the reflection and transmission phenomena of the seismic and EM wavefields at the interfaces of contrasting media. This may allow for the identification of abnormal locations, thus highlighting the capability of seismoelectric response simulation to detect subsurface properties.

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