IMPROVED ACCURACY FOR LOCALLY ONE-DIMENSIONAL METHODS FOR PARABOLIC EQUATIONS

2001 ◽  
Vol 11 (09) ◽  
pp. 1563-1579 ◽  
Author(s):  
JIM DOUGLAS ◽  
SEONGJAI KIM
Keyword(s):  
Parabolic Equations ◽  
Wave Equations ◽  
Computational Cost ◽  
Implicit Time ◽  
Differential Problem ◽  
Time Stepping ◽  
Backward Euler ◽  
The Cost ◽  
Improved Accuracy ◽  
Crank Nicolson

Classical alternating direction (AD) and fractional step (FS) methods for parabolic equations, based on some standard implicit time-stepping procedure such as Crank–Nicolson, can have errors associated with the AD or FS perturbations that are much larger than the errors associated with the underlying time-stepping procedure. We show that minor modifications in the AD and FS procedures can virtually eliminate the perturbation errors at an additional computational cost that is less than 10% of the cost of the original AD or FS method. Moreover, after these modifications, the AD and FS procedures produce identical approximations of the solution of the differential problem. It is also shown that the same perturbation of the Crank–Nicolson procedure can be obtained with AD and FS methods associated with the backward Euler time-stepping scheme. An application of the same concept is presented for second-order wave equations.

Numerical simulation of higher-order diffusion-wave equations of variable coefficients using the meshless spectral method

2020 ◽  
pp. 2150010
Author(s):  
Manzoor Hussain ◽  
Sirajul Haq
Keyword(s):  
Wave Equations ◽  
Interpolation Method ◽  
Variable Coefficients ◽  
Higher Order ◽  
Implicit Time ◽  
Test Problems ◽  
Time Step ◽  
Diffusion Wave ◽  
Time Step Size ◽  
Time Stepping

In this paper, meshless spectral interpolation technique using implicit time stepping scheme is proposed for the numerical simulations of time-fractional higher-order diffusion wave equations (TFHODWEs) of variable coefficients. Meshless shape functions, obtained from radial basis functions (RBFs) and point interpolation method (PIM), are used for spatial approximation. Central differences coupled with quadrature rule of [Formula: see text] are employed for fractional temporal approximation. For advancement of solution, an implicit time stepping scheme is then invoked. Simulations performed for different benchmark test problems feature good agreement with exact solutions. Stability analysis of the proposed method is theoretically discussed and computationally validated to support the analysis. Accuracy and efficiency of the proposed method are assessed via [Formula: see text], [Formula: see text] and [Formula: see text] error norms as well as number of nodes [Formula: see text] and time step-size [Formula: see text].

scholarly journals Efficient estimation of time-mean states of ocean models using 4D-Var and implicit time-stepping

2007 ◽  
Vol 14 (6) ◽  
pp. 777-788
Author(s):  
A. D. Terwisscha van Scheltinga ◽  
H. A. Dijkstra
Keyword(s):  
Ocean Circulation ◽  
Signal To Noise Ratio ◽  
Singular Spectrum Analysis ◽  
Computational Cost ◽  
Ocean Model ◽  
Efficient Estimation ◽  
Implicit Time ◽  
Time Stepping ◽  
High Frequency Variability ◽  
Mean State

Abstract. We propose an efficient method for estimating a time-mean state of an ocean model subject to given observations using implicit time-stepping. The new method uses (i) an implicit implementation of the 4D-Var method to fit the model trajectory to the observations, and (ii) a pre-processor which applies a multi-channel singular spectrum analysis to enhance the signal-to-noise ratio of the observational data and to filter out the high frequency variability. This approach enables one to estimate the time-mean model state using larger time-steps than is possible with an explicit model. The performance of the method is presented for two test cases within a barotropic quasi-geostrophic nonlinear model of the wind-driven double-gyre ocean circulation. The method turns out to be accurate and, in comparison with the time-mean state computed with an explicit version of the model, relatively cheap in computational cost.

scholarly journals Fiber-based modeling and simulation of skeletal muscles

2021 ◽  
Vol 52 (1) ◽  
pp. 1-30
Author(s):  
M. H. Gfrerer ◽  
B. Simeon
Keyword(s):  
Continuum Mechanics ◽  
Skeletal Muscles ◽  
Computational Cost ◽  
Rigid Bodies ◽  
Muscle Model ◽  
Implicit Time ◽  
Cable Model ◽  
One Dimensional ◽  
Time Stepping ◽  
The Hill

AbstractThis paper presents a novel fiber-based muscle model for the forward dynamics of the musculoskeletal system. While bones are represented by rigid bodies, the muscles are taken into account by means of one-dimensional cables that obey the laws of continuum mechanics. In contrast to standard force elements such as the Hill-type muscle model, this approach is close to the real physiology and also avoids the issue of wobbling masses. On the other hand, the computational cost is rather low in comparison with full 3D continuum mechanics simulations. The cable model includes sliding contact between individual fibers as well as between fibers and bones. For the discretization, cubic finite elements are employed in combination with implicit time stepping. Several validation studies and the simulation of a motion scenario for the upper limb demonstrate the potential of the fiber-based approach.

scholarly journals Full-waveform inversion using seislet regularization

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. A43-A49 ◽  
Author(s):  
Zhiguang Xue ◽  
Hejun Zhu ◽  
Sergey Fomel
Keyword(s):  
Wave Equations ◽  
Structural Information ◽  
Random Noise ◽  
Waveform Inversion ◽  
Computational Cost ◽  
Optimization Approach ◽  
Full Waveform Inversion ◽  
Velocity Models ◽  
Full Waveform ◽  
The Cost

Because of inaccurate, incomplete, and inconsistent waveform records, full-waveform inversion (FWI) in the framework of a local optimization approach may not have a unique solution, and thus it remains an ill-posed inverse problem. To improve the robustness of FWI, we have developed a new model regularization approach that enforced the sparsity of solutions in the seislet domain. The construction of seislet basis functions requires structural information that can be estimated iteratively from migration images. We implement FWI with seislet regularization using nonlinear shaping regularization and impose sparseness by applying soft thresholding on the updated model in the seislet domain at each iteration of the data-fitting process. The main extra computational cost of the method relative to standard FWI is the cost of applying forward and inverse seislet transforms at each iteration. This cost is almost negligible compared with the cost of solving wave equations. Numerical tests using the synthetic Marmousi model demonstrate that seislet regularization can greatly improve the robustness of FWI by recovering high-resolution velocity models, particularly in the presence of strong crosstalk artifacts from simultaneous sources or strong random noise in the data.

Long Time Steps for Advection: MPDATA with implicit time stepping

2021 ◽  
Author(s):  
Hilary Weller ◽  
James Woodfield ◽  
Christian Kuehnlein
Keyword(s):  
Implicit Time ◽  
Courant Number ◽  
Time Stepping ◽  
Phase Errors ◽  
Advection Schemes ◽  
The Matrix ◽  
Long Time ◽  
Strong Winds ◽  
Lagrangian Advection ◽  
The Cost

<p>Semi-Lagrangian advection schemes are accurate and efficient and retain accuracy and stability even for large Courant numbers but are not conservative. Flux-form semi-Lagrangian is conservative and in principle can be used to achieve large Courant numbers. However this is complicated and would be prohibitively expensive on grids that are not logically rectangular. </p><p>Strong winds or updrafts can lead to localised violations of Courant number restrictions which can cause a model with explicit Eulerian advection to crash. Schemes are needed that remain stable in the presence of large Courant numbers. However accuracy in the presence of localised large Courant numbers may not be so crucial.</p><p>Implicit time stepping for advection is not popular in atmospheric science because of the cost of the global matrix solution and the phase errors for large Courant numbers. However implicit advection is simple to implement (once appropriate matrix solvers are available) and is conservative on any grid structure and can exploit improvements in solver efficiency and parallelisation. This talk will describe an implicit version of the MPDATA advection scheme and show results of linear advection test cases. To optimise accuracy and efficiency, implicit time stepping is only used locally where needed. This makes the matrix inversion problem local rather than global. With implicit time stepping MPDATA retains positivity, smooth solutions and accuracy in space and time.</p>

Planned seismic imaging using explicit one-way operators

Geophysics ◽  
2005 ◽  
Vol 70 (5) ◽  
pp. S101-S109 ◽  
Author(s):  
Robert J. Ferguson ◽  
Gary F. Margrave
Keyword(s):  
Fourier Transform ◽  
Seismic Imaging ◽  
Computational Cost ◽  
Velocity Model ◽  
Fixed Cost ◽  
Positioning Error ◽  
Model Accuracy ◽  
The Cost ◽  
Improved Accuracy ◽  
Lateral Positioning

A method is presented to compare the accuracy and computational cost of explicit one-way extrapolation operators as used in seismic imaging. For a given model, accuracy is measured in terms of lateral positioning error, and cost is calculated relative to the cost of the spatial fast Fourier transform. The result is a planned imaging scheme that achieves the greatest accuracy with respect to the velocity model for a fixed cost. To demonstrate, we use a 2D section of the EAGE/SEG salt model and assemble a suite of the most common operators. The data are imaged using each operator individually, and the results are compared to the result from the plan-based algorithm. The planned image is shown to return improved accuracy for no additional cost.

Equivalent accuracy at a fraction of the cost: Overcoming temporal dispersion

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. T189-T196 ◽  
Author(s):  
Yunyue Elita Li ◽  
Mandy Wong ◽  
Robert Clapp
Keyword(s):  
Computational Cost ◽  
Numerical Dispersion ◽  
Propagation Time ◽  
Time Stepping ◽  
Time Dispersion ◽  
Input Waveform ◽  
Higher Order Modeling ◽  
The Cost ◽  
Inversion Techniques ◽  
And Inversion

Numerical dispersion in finite-difference (FD) modeling produces coherent artifacts, severely constraining the resolution of advanced imaging and inversion techniques. Conventionally, numerical dispersion is reduced by increasing the order of accuracy of the FD operators, and we resign ourselves to paying the high computational cost that is incurred. Assuming no spatial dispersion, we have found that FD time dispersion is independent of the medium velocity and the spatial grid for propagation, and only depends on the time-stepping scheme and the propagation time. Based on this observation, we have devised postpropagation filters to collapse the time-dispersion effect of FD modeling. Our dispersion correction filters are designed by comparing the input waveform with dispersive waveforms obtained by 1D forward modeling. These filters are then applied on multidimensional shot records to eliminate the time dispersion by two schemes: (1) stationary filtering plus interpolation and (2) nonstationary filtering. We have found with 1D and 2D examples that the time dispersion is effectively removed by our postpropagation filtering at a negligible cost compared with a higher order modeling scheme.

Nearly perfectly matched layer absorber for viscoelastic wave equations

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. T335-T345
Author(s):  
Enjiang Wang ◽  
José M. Carcione ◽  
Jing Ba ◽  
Mamdoh Alajmi ◽  
Ayman N. Qadrouh
Keyword(s):  
Wave Equations ◽  
Pseudospectral Method ◽  
Perfectly Matched Layer ◽  
Zener Model ◽  
Fourier Pseudospectral Method ◽  
Time Stepping ◽  
First Case ◽  
Stress Strain Relation ◽  
Viscoelastic Wave ◽  
Spatial Derivatives

We have applied the nearly perfectly matched layer (N-PML) absorber to the viscoelastic wave equation based on the Kelvin-Voigt and Zener constitutive equations. In the first case, the stress-strain relation has the advantage of not requiring additional physical field (memory) variables, whereas the Zener model is more adapted to describe the behavior of rocks subject to wave propagation in the whole frequency range. In both cases, eight N-PML artificial memory variables are required in the absorbing strips. The modeling simulates 2D waves by using two different approaches to compute the spatial derivatives, generating different artifacts from the boundaries, namely, 16th-order finite differences, where reflections from the boundaries are expected, and the staggered Fourier pseudospectral method, where wraparound occurs. The time stepping in both cases is a staggered second-order finite-difference scheme. Numerical experiments demonstrate that the N-PML has a similar performance as in the lossless case. Comparisons with other approaches (S-PML and C-PML) are carried out for several models, which indicate the advantages and drawbacks of the N-PML absorber in the anelastic case.

scholarly journals A Finite Element Splitting Method for a Convection-Diffusion Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 717-725 ◽  
Author(s):  
Vidar Thomée
Keyword(s):  
Finite Element ◽  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Euler Approximation ◽  
Convection Diffusion ◽  
Time Stepping ◽  
Backward Euler ◽  
Spatially Periodic ◽  
Forward Euler

AbstractFor a spatially periodic convection-diffusion problem, we analyze a time stepping method based on Lie splitting of a spatially semidiscrete finite element solution on time steps of length k, using the backward Euler method for the diffusion part and a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {k/m} for the convection part. This complements earlier work on time splitting of the problem in a finite difference context.

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