Time step n-tupling for wave equations

Geophysics ◽  
2017 ◽  
Vol 82 (6) ◽  
pp. T249-T254 ◽  
Author(s):  
Lasse Amundsen ◽  
Ørjan Pedersen
Keyword(s):  
Discrete Time ◽  
Wave Equations ◽  
Fourier Method ◽  
Computational Effort ◽  
Second Order ◽  
Time Step ◽  
Time Stepping ◽  
Previous Time ◽  
Time Integrators ◽  
New Time

We have constructed novel temporal discretizations for wave equations. We first select an explicit time integrator that is of second order, leading to classic time marching schemes in which the next value of the wavefield at the discrete time [Formula: see text] is computed from current values known at time [Formula: see text] and the previous time [Formula: see text]. Then, we determine how the time step can be doubled, tripled, or generally, [Formula: see text]-tupled, producing a new time-stepping method in which the next value of the wavefield at the discrete time [Formula: see text] is computed from current values known at time [Formula: see text] and the previous time [Formula: see text]. In-between time values of the wavefield are eliminated. Using the Fourier method to calculate space derivatives, the new time integrators allow larger stable time steps than traditional time integrators; however, like the Lax-Wendroff procedure, they require more computational effort per time step. Because the new schemes are developed from the classic second-order time-stepping scheme, they will have the same properties, except the Courant-Friedrichs-Lewy stability condition, which becomes relaxed by the factor [Formula: see text] compared with the classic scheme. As an example, we determine the method for solving scalar wave propagation in which doubling the time step is 15% faster than a Lax-Wendroff correction scheme of the same spatial order because it can increase the time step by [Formula: see text] only.

scholarly journals Construction and convergence analysis of conservative second order local time discretisation for linear wave equations

2021 ◽  
Author(s):  
Juliette Chabassier ◽  
Sébastien Imperiale
Keyword(s):  
Local Time ◽  
Wave Equations ◽  
Second Order ◽  
Linear Wave ◽  
Time Step ◽  
Time Stepping ◽  
Time Discretisation ◽  
Regularity Properties ◽  
Local Time Stepping ◽  
Linear Wave Equations

In this work we present and analyse a time discretisation strategy for linear wave equations that aims at using locally in space the most adapted time discretisation among a family of implicit or explicit centered second order schemes. The proposed family of schemes is adapted to domain decomposition methods such as the mortar element method. They correspond in that case to local implicit schemes and to local time stepping. We show that, if some regularity properties of the solution are satisfied and if the time step verifies a stability condition, then the family of proposed time discretisations provides, in a strong norm, second order space-time convergence. Finally, we provide 1D and 2D numerical illustrations that confirm the obtained theoretical results and we compare our approach on 1D test cases to other existing local time stepping strategies for wave equations.

Time-space-domain implicit finite-difference methods for modeling acoustic wave equations

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. T175-T193 ◽  
Author(s):  
Enjiang Wang ◽  
Jing Ba ◽  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Wave Equations ◽  
Finite Difference Methods ◽  
Second Order ◽  
Computational Time ◽  
Time Step ◽  
Space Domain ◽  
Time Space ◽  
Temporal Dispersion

It has been proved that the implicit spatial finite-difference (FD) method can obtain higher accuracy than explicit FD by using an even smaller operator length. However, when only second-order FD in time is used, the combined FD scheme is prone to temporal dispersion and easily becomes unstable when a relatively large time step is used. The time-space domain FD can suppress the temporal dispersion. However, because the spatial derivatives are solved explicitly, the method suffers from spatial dispersion and a large spatial operator length has to be adopted. We have developed two effective time-space-domain implicit FD methods for modeling 2D and 3D acoustic wave equations. First, the high-order FD is incorporated into the discretization for the second-order temporal derivative, and it is combined with the implicit spatial FD. The plane-wave analysis method is used to derive the time-space-domain dispersion relations, and two novel methods are proposed to determine the spatial and temporal FD coefficients in the joint time-space domain. First, we fix the implicit spatial FD coefficients and derive the quadratic convex objective function with respect to temporal FD coefficients. The optimal temporal FD coefficients are obtained by using the linear least-squares method. After obtaining the temporal FD coefficients, the SolvOpt nonlinear algorithm is applied to solve the nonquadratic optimization problem and obtain the optimized temporal and spatial FD coefficients simultaneously. The dispersion analysis, stability analysis, and modeling examples validate that the proposed schemes effectively increase the modeling accuracy and improve the stability conditions of the traditional implicit schemes. The computational efficiency is increased because the schemes can adopt larger time steps with little loss of spatial accuracy. To reduce the memory requirement and computational time for storing and calculating the FD coefficients, we have developed the representative velocity strategy, which only computes and stores the FD coefficients at several selected velocities. The modeling result of the 2D complicated model proves that the representative velocity strategy effectively reduces the memory requirements and computational time without decreasing the accuracy significantly when a proper velocity interval is used.

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].

A spectral scheme for wave propagation simulation in 3-D elastic‐anisotropic media

Geophysics ◽  
1992 ◽  
Vol 57 (12) ◽  
pp. 1593-1607 ◽  
Author(s):  
José M. Carcione ◽  
Dan Kosloff ◽  
Alfred Behle ◽  
Geza Seriani
Keyword(s):  
Wave Propagation ◽  
Fourier Method ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Computational Effort ◽  
Second Order ◽  
Rapid Expansion ◽  
Integration Algorithm ◽  
Fourier Pseudospectral Method ◽  
Band Limited

This work presents a new scheme for wave propagation simulation in three‐dimensional (3-D) elastic-anisotropic media. The algorithm is based on the rapid expansion method (REM) as a time integration algorithm, and the Fourier pseudospectral method for computation of the spatial derivatives. The REM expands the evolution operator of the second‐order wave equation in terms of Chebychev polynomials, constituting an optimal series expansion with exponential convergence. The modeling allows arbitrary elastic coefficients and density in lateral and vertical directions. Numerical methods which are based on finite‐difference techniques (in time and space) are not efficient when applied to realistic 3-D models, since they require considerable computer memory and time to obtain accurate results. On the other hand, the Fourier method permits a significant reduction of the working space, and the REM algorithm gives machine accuracy with half the computational effort as the usual second-order temporal differencing scheme. The new algorithm provides spectral accuracy for band limited wave propagation with no temporal or spatial dispersion. Hence, the combination REM/Fourier method could be considered at present the fastest and the most accurate of all the algorithms based on spectral methods in terms of efficiency of computer time. The technique is successfully tested with the analytical solution in the symmetry axis of a 3-D homogeneous transversely isotropic solid. Computed radiation patterns in clay shale and sandstone show the characteristics predicted by the theory. A final example computes synthetic seismograms showing the effects of shear‐wave splitting of a model where an azimuthally anisotropic cracked shale layer is inside a transversely isotropic sandstone.

scholarly journals A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems

2019 ◽  
Vol 19 (3) ◽  
pp. 681-701 ◽  
Author(s):  
Max Gunzburger ◽  
Nan Jiang ◽  
Zhu Wang
Keyword(s):  
Stokes Equations ◽  
Second Order ◽  
Navier Stokes ◽  
Conditional Stability ◽  
Initial Condition ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Flow Simulations ◽  
Flow Problems ◽  
Time Stepping

AbstractWe consider settings for which one needs to perform multiple flow simulations based on the Navier–Stokes equations, each having different initial condition data, boundary condition data, forcing functions, and/or coefficients such as the viscosity. For such settings, we propose a second-order time accurate ensemble-based method that to simulate the whole set of solutions, requires, at each time step, the solution of only a single linear system with multiple right-hand-side vectors. Rigorous analyses are given proving the conditional stability and establishing error estimates for the proposed algorithm. Numerical experiments are provided that illustrate the analyses.

scholarly journals Artificial Compressibility Methods for the Incompressible Navier–Stokes Equations Using Lowest-Order Face-Based Schemes on Polytopal Meshes

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Riccardo Milani ◽  
Jérôme Bonelle ◽  
Alexandre Ern
Keyword(s):  
Stokes Equations ◽  
Time Discretization ◽  
Second Order ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Artificial Compressibility ◽  
Time Stepping ◽  
Monolithic Approach ◽  
Polytopal Meshes

Abstract We investigate artificial compressibility (AC) techniques for the time discretization of the incompressible Navier–Stokes equations. The space discretization is based on a lowest-order face-based scheme supporting polytopal meshes, namely discrete velocities are attached to the mesh faces and cells, whereas discrete pressures are attached to the mesh cells. This face-based scheme can be embedded into the framework of hybrid mixed mimetic schemes and gradient schemes, and has close links to the lowest-order version of hybrid high-order methods devised for the steady incompressible Navier–Stokes equations. The AC time-stepping uncouples at each time step the velocity update from the pressure update. The performances of this approach are compared against those of the more traditional monolithic approach which maintains the velocity-pressure coupling at each time step. We consider both first-order and second-order time schemes and either an implicit or an explicit treatment of the nonlinear convection term. We investigate numerically the CFL stability restriction resulting from an explicit treatment, both on Cartesian and polytopal meshes. Finally, numerical tests on large 3D polytopal meshes highlight the efficiency of the AC approach and the benefits of using second-order schemes whenever accurate discrete solutions are to be attained.

Three‐dimensional acoustic modeling by the Fourier method

Geophysics ◽  
1988 ◽  
Vol 53 (9) ◽  
pp. 1175-1183 ◽  
Author(s):  
Moshe Reshef ◽  
Dan Kosloff ◽  
Mickey Edwards ◽  
Chris Hsiung
Keyword(s):  
Three Dimensional ◽  
Fourier Method ◽  
Forward Modeling ◽  
Secondary Memory ◽  
Storage Device ◽  
Data Sets ◽  
Time Step ◽  
Absorbing Boundary ◽  
Time Stepping ◽  
Spatial Grid

A three‐dimensional forward modeling algorithm, allowing arbitrary density and arbitrary wave propagation velocity in lateral and vertical directions, directly solves the acoustic wave equation through spatial and temporal discretization. Spatial partial differentiation is performed in the Fourier domain. Time stepping is performed with a second‐order differencing operator. Modeling includes an optional free surface above the spatial grid. An absorbing boundary is applied on the lateral and bottom edges of the spatial grid. Three‐dimensional forward modeling represents a challenge for computer technology. Computation of meaningfully sized models requires extensive calculations and large three‐dimensional data sets which must be retrieved and restored during the computation of each time step. The computational feasibility of the Fourier method is demonstrated by implementation on the multiprocessor CRAY X‐MP computer system using the large secondary memory of the solid‐state storage device (SSD). Calculations use vectorization and parallel processing architecture. The similarity of numerical and analytical results indicates sufficient accuracy for many applications.

On Direct and Semi-Direct Inverse of Stokes, Helmholtz and Laplacian Operators in View of Time-Stepper-Based Newton and Arnoldi Solvers in Incompressible CFD

2013 ◽  
Vol 14 (4) ◽  
pp. 1103-1119 ◽  
Author(s):  
H. Vitoshkin ◽  
A. Yu. Gelfgat
Keyword(s):  
Three Dimensional ◽  
Reynolds Numbers ◽  
Computational Effort ◽  
Stokes Operator ◽  
Time Step ◽  
Time Stepping ◽  
Fast Direct Solvers ◽  
Large Time Step ◽  
Helmholtz Operators ◽  
Laplacian Operators

AbstractFactorization of the incompressible Stokes operator linking pressure and velocity is revisited. The main purpose is to use the inverse of the Stokes operator with a large time step as a preconditioner for Newton and Arnoldi iterations applied to computation of steady three-dimensional flows and study of their stability. It is shown that the Stokes operator can be inversed within an acceptable computational effort. This inverse includes fast direct inverses of several Helmholtz operators and iterative inverse of the pressure matrix. It is shown, additionally, that fast direct solvers can be attractive for the inverse of the Helmholtz and Laplace operators on fine grids and at large Reynolds numbers, as well as for other problems where convergence of iterative methods slows down. Implementation of the Stokes operator inverse to time-stepping-based formulation of the Newton and Arnoldi iterations is discussed.

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