A New Class of Highly Accurate Solvers for Ordinary Differential Equations

Gläser, Andreas; Rokhlin, Vladimir

doi:10.1007/s10915-008-9245-1

Cited by 46 publications

(64 citation statements)

References 16 publications

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“…Differentiation matrices that incorporate other boundary conditions, such as the Neumann boundary condition: 26) are similarly constructed by modifying the functions µ j in (3.15). Likewise, differentiation matrices for computing first derivatives are constructed with the straightforward modification of the above scheme.…”

Section: Differentiation Matrices Incorporating Boundary Conditionsmentioning

confidence: 99%

A New Class of Highly Accurate Differentiation Schemes Based on the Prolate Spheroidal Wave Functions

Kong¹,

Rokhlin²

2011

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We introduce a new class of numerical differentiation schemes constructed for the efficient solution of time-dependent PDEs that arise in wave phenomena. The schemes are constructed via the prolate spheroidal wave functions (PSWFs). Compared to existing differentiation schemes based on orthogonal polynomials, the new class of differentiation schemes requires fewer points per wavelength to achieve the same accuracy when it is used to approximate derivatives of bandlimited functions. In addition, the resulting differentiation matrices have spectral radii that grow asymptotically as m for the case of first derivatives, and m 2 for second derivatives, with m being the dimensions of the matrices. The above results mean that the new class of differentiation schemes is more efficient in the solution of time-dependent PDEs compared to existing schemes such as the Chebyshev collocation method. The improvements are particularly prominent in large-scale time-dependent PDEs, in which the solutions contain large numbers of wavelengths in the computational domains.

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Section: Differentiation Matrices Incorporating Boundary Conditionsmentioning

confidence: 99%

A New Class of Highly Accurate Differentiation Schemes Based on the Prolate Spheroidal Wave Functions

Kong¹,

Rokhlin²

2011

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“…For a fixed order ν, we use the method found in [9] to numerically integrate differential equation (2.6) to evaluate Bessel functions of the first kind. For any λ ∈ C, after a change of variable equation (2.6) becomes…”

Section: Bessel Functionsmentioning

confidence: 99%

“…. , x n can be numerically calculated using the method contained in [9]; once calculated, α k is given by the formula…”

Section: Hermite Polynomialsmentioning

confidence: 99%

“…. , x n , which are used in formulas (2.122-2.125), can be numerically calculated using the method contained in [9]. The coefficients in expansion (2.121) are then given by the formula…”

Section: Laguerre Polynomialsmentioning

confidence: 99%

“…, α n−1 is a well-conditioned procedure. Using the methods of [9], we can numerically calculate the roots of any PSWF as well as evaluate any PSWF on the interval [−1, 1]. Suppose now that the coefficients in expansion (2.140) are known, and one wishes to evaluate the expansion at the n points x 1 , x 2 , .…”

Section: Prolate Spheroidal Wave Functionsmentioning

confidence: 99%

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A New Class of Analysis-Based Fast Transforms

Rokhlin¹,

O’Neil²

2007

Self Cite

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We introduce a new approach to the rapid numerical application to arbitrary vectors of certain types of linear operators. Inter alia, our scheme is applicable to many classical integral transforms, and to the expansions associated with most families of classical special functions; among the latter are Bessel functions, Legendre, Hermite, and Laguerre polynomials, Spherical Harmonics, Prolate Spheroidal Wave functions, and a number of others. In all these cases, the CPU time requirements of our algorithm are of the order O(n log n), where n is the size of the matrix to be applied. The performance of our algorithm is illustrated via a number of numerical examples.A new class of analysis-based fast transforms

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Explicit Marching‐on‐in‐time Solvers for Second‐kind Time Domain Integral Equations

Chen¹,

Sayed²,

Ülkü³

et al. 2022

Advances in Time‐Domain Computational Electromagnetic Methods

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Time domain methods are often preferred over their frequency domain counterparts for numerical electromagnetic analysis since they can produce broadband data from a single simulation, can account for nonlinearities directly, and often provide immediate physical interpretation of the problem's solution. Among the time domain methods, time domain integral equation (TDIE) solvers have recently found widespread use and they have become an attractive alternative to differential equation solvers such 1 as finite difference time domain schemes and time domain finite element method, especially for open region scattering problems. This is because TDIE solvers do not suffer from numerical phase dispersion, do not require approximate radiation boundary conditions (such as absorbing boundary conditions and perfectly matched layers), and their time step size is not restricted by a Courant-Friedrichs-Lewy-like condition. Almost all traditional TDIE solvers utilize an implicit time marching scheme, i.e., they call for solution of a (sparse) matrix at every iteration. This reduces the computational efficiency under low-frequency excitations and makes the use of TDIE solvers in multiphysics frameworks a challenge. To address these challenges, recently, various explicit marching-on-in-time (MOT) schemes have been developed to solve the second kind TDIEs for perfect electrically conducting and dielectric scatterers. This chapter details the formulation and implementation of these TDIE solvers. These schemes cast the second kind TDIE in the form of an ordinary differential equation (ODE).A classical scheme such as those making use of Rao-Wilton-Glisson and Schubert-Wilton-Glisson basis functions and Nyström integration is used for spatial discretization. Then, a predictor-corrector scheme is used to integrate the spatially discretized ODE systems in time for the expansion coefficients of the unknowns. The explicit TDIE solvers described in this chapter use the same time step size as their implicit counterparts without sacrificing from accuracy and stability of the solution. In addition, they are significantly faster under low-frequency excitations. Numerical results are presented to demonstrate these computational benefits.

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A New Class of Highly Accurate Solvers for Ordinary Differential Equations

Cited by 46 publications

References 16 publications

A New Class of Highly Accurate Differentiation Schemes Based on the Prolate Spheroidal Wave Functions

A New Class of Highly Accurate Differentiation Schemes Based on the Prolate Spheroidal Wave Functions

A New Class of Analysis-Based Fast Transforms

Explicit Marching‐on‐in‐time Solvers for Second‐kind Time Domain Integral Equations

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