An adaptive wavelet collocation method for the solution of partial differential equations on the sphere

Mehra, Mani; Kevlahan, Nicholas

doi:10.1016/j.jcp.2008.02.004

Cited by 57 publications

(51 citation statements)

References 32 publications

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“…The elliptic solver can be used for the PDE constraints in evolution problems, such as the Poisson equation in the pressure correction method for the incompressible Navier-Stokes equations. Together, our previous article [16] and the present paper thus provide a complete set of tools for the efficient and accurate adaptive solution of PDEs on the sphere. In particular, we can now solve the full incompressible Navier-Stokes equations on the sphere, taking advantage of wavelet multilevel decomposition and compression.…”

mentioning

confidence: 84%

“…We first review the calculation of the Laplace-Beltrami operator on a standard spherical geodesic grid as described in [16]. For any point p on the surface of S, it is known that [26] (2.5)…”

Section: 2mentioning

confidence: 99%

“…where the coefficients s j k,m can be chosen according as in [16]. The procedure for finding Laplacian at all grid points is given in Algorithm 2.…”

Section: 4mentioning

confidence: 99%

“…In a companion paper [16], we developed an adaptive wavelet collocation method for the representation of the Laplace-Beltrami, Jacobian, and flux divergence operators on the sphere. We also demonstrated the accuracy and efficiency of this approach for the adaptive solution of the advection and diffusion equations on the sphere.…”

mentioning

confidence: 99%

“…Wavelets on the sphere. We use the spherical wavelet multiresolution analysis, derivative approximation, and grid adaptation strategy described in [16].…”

mentioning

confidence: 99%

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An Adaptive Multilevel Wavelet Solver for Elliptic Equations on an Optimal Spherical Geodesic Grid

Mehra¹,

Kevlahan²

2008

SIAM J. Sci. Comput.

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Abstract. An adaptive multilevel wavelet solver for elliptic equations on an optimal spherical geodesic grid is developed. The method is based on second-generation spherical wavelets on almost uniform optimal spherical geodesic grids. It is an extension of the adaptive multilevel wavelet solver [O. V. Vasilyev and N. K.-R. Kevlahan, J. Comput. Phys., 206 (2005), pp. 412-431] to curved manifolds. Wavelet decomposition is used for grid adaption and interpolation. A hierarchical finite difference scheme based on the wavelet multilevel decomposition is used to approximate the LaplaceBeltrami operator. The optimal spherical geodesic grid [Internat. J. Comput. Geom. Appl., 16 (2006), pp. 75-93] is convergent in terms of local mean curvature and has lower truncation error than conventional spherical geodesic grids. The overall computational complexity of the solver is O(N ), where N is the number of grid points after adaptivity. The accuracy and efficiency of the method is demonstrated for the spherical Poisson equation. Although the present paper considers the sphere, the strength of this new method is that it can be extended easily to other curved manifolds by choosing an appropriate coarse approximation and using recursive surface subdivision.

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mentioning

confidence: 84%

Section: 2mentioning

confidence: 99%

“…where the coefficients s j k,m can be chosen according as in [16]. The procedure for finding Laplacian at all grid points is given in Algorithm 2.…”

Section: 4mentioning

confidence: 99%

mentioning

confidence: 99%

“…Wavelets on the sphere. We use the spherical wavelet multiresolution analysis, derivative approximation, and grid adaptation strategy described in [16].…”

mentioning

confidence: 99%

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An Adaptive Multilevel Wavelet Solver for Elliptic Equations on an Optimal Spherical Geodesic Grid

Mehra¹,

Kevlahan²

2008

SIAM J. Sci. Comput.

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The construction of finite element multiwavelets for adaptive structural analysis

Wang

Chen

et al. 2011

Int. J. Numer. Meth. Biomed. Engng.

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SUMMARYA design method of finite element multiwavelets is proposed for adaptive analysis of structural problems. A multiresolution analysis for Lagrange and Hermite finite element space is discussed. New classes of finite element multiwavelets are constructed by the lifting scheme according to the operators of structural problems. Compared with classical wavelet methods, the finite element multiwavelet method is more flexible and robust for multiscale structural analysis. Based on the operator-orthogonality of the finite element multiwavelets, we propose a new adaptive scheme for the finite element multiwavelet method by adding new multiwavelets into the domain where the error estimator is larger than a given threshold value. Numerical examples demonstrate that the finite element multiwavelets are flexible and are efficient bases in solving structural problems.

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A computational methodology for two‐dimensional fluid flows

Alam

Walsh

Hossain

et al. 2014

Numerical Methods in Fluids

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SUMMARYA weighted residual collocation methodology for simulating two‐dimensional shear‐driven and natural convection flows has been presented. Using a dyadic mesh refinement, the methodology generates a basis and a multiresolution scheme to approximate a fluid flow. To extend the benefits of the dyadic mesh refinement approach to the field of computational fluid dynamics, this article has studied an iterative interpolation scheme for the construction and differentiation of a basis function in a two‐dimensional mesh that is a finite collection of rectangular elements. We have verified that, on a given mesh, the discretization error is controlled by the order of the basis function. The potential of this novel technique has been demonstrated with some representative examples of the Poisson equation. We have also verified the technique with a dynamical core of a two‐dimensional flow in primitive variables. An excellent result has been observed—on resolving a shear layer and on the conservation of the potential and the kinetic energies—with respect to previously reported benchmark simulations. In particular, the shear‐driven simulation at CFL = 2.5 (Courant–Friedrichs–Lewy) and scriptRe MathClass-rel= 1000 (Reynolds number) exhibits a linear speed up of CPU time with an increase of the time step, Δt. For the natural convection flow, the conversion of the potential energy to the kinetic energy and the conservation of total energy is resolved by the proposed method. The computed streamlines and the velocity fields have been demonstrated. Copyright © 2014 John Wiley & Sons, Ltd.

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An adaptive wavelet collocation method for the solution of partial differential equations on the sphere

Cited by 57 publications

References 32 publications

An Adaptive Multilevel Wavelet Solver for Elliptic Equations on an Optimal Spherical Geodesic Grid

An Adaptive Multilevel Wavelet Solver for Elliptic Equations on an Optimal Spherical Geodesic Grid

The construction of finite element multiwavelets for adaptive structural analysis

A computational methodology for two‐dimensional fluid flows

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