Accuracy, Resolution and Stability Properties of a Modified Chebyshev Method

Jodi Mead; Rosemary A. Renaut

Accuracy, Resolution and Stability Properties of a Modified Chebyshev Method

Jodi Mead, Rosemary A. Renaut

Mathematics Department

Arizona State University, Tempe Campus

Research output: Contribution to journal › Article › peer-review

Abstract

While the Chebyshev pseudospectral method provides a spectrally accurate method, integration of partial differential equations with spatial derivatives of order M requires time steps of approximately O(N −2M ) for stable explicit solvers. Theoretically, time steps may be increased to O ( N −M ) with the use of a parameter, α-dependent mapped method introduced by Kosloff and Tal-Ezer [ J. Comput. Phys. , 104 (1993), pp. 457–469]. Our analysis focuses on the utilization of this method for reasonable practical choices for N , namely N &lsim; 30, as may be needed for two- or three dimensional modeling. Results presented confirm that spectral accuracy with increasing N is possible both for constant α (Hesthaven, Dinesen, and Lynov [ J. Comput. Phys. , 155 (1999), pp. 287–306]) and for α scaled with N , α sufficiently different from 1 (Don and Solomonoff [ SIAM J. Sci. Comput. , 18 (1997), pp. 1040–1055]). Theoretical bounds, however, show that any realistic choice for α, in which both resolution and accuracy considerations are imposed, permits no more than a doubling of the time step fora stable explicit integrator in time, much less than the O ( N ) improvement claimed by Kosloff and Tal-Ezer. On the other hand, by choosing α carefully, it is possible to improve on the resolution of the Chebyshev method; in particular, one may achieve satisfactory resolution with fewer than π points per wavelength. Moreover, this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about 8 points per wavelength. Our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one-dimensional wave equations. Specifically, while α can be chosen such that the mapped method improves the accuracy and resolution of the Chebyshev method, for practical choices of N , it is not possible to achieve both single precision accuracy and gain the advantage of an O ( N −M ) time step.

Original language	American English
Journal	SIAM Journal on Scientific Computing
State	Published - 1 Jan 2002

Keywords

Chebyshev collocation
accuracy
partial differential equations
stability

EGS Disciplines

Mathematics

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title = "Accuracy, Resolution and Stability Properties of a Modified Chebyshev Method",

abstract = " While the Chebyshev pseudospectral method provides a spectrally accurate method, integration of partial differential equations with spatial derivatives of order M requires time steps of approximately O(N −2M ) for stable explicit solvers. Theoretically, time steps may be increased to O ( N −M ) with the use of a parameter, α-dependent mapped method introduced by Kosloff and Tal-Ezer [ J. Comput. Phys. , 104 (1993), pp. 457–469]. Our analysis focuses on the utilization of this method for reasonable practical choices for N , namely N &lsim; 30, as may be needed for two- or three dimensional modeling. Results presented confirm that spectral accuracy with increasing N is possible both for constant α (Hesthaven, Dinesen, and Lynov [ J. Comput. Phys. , 155 (1999), pp. 287–306]) and for α scaled with N , α sufficiently different from 1 (Don and Solomonoff [ SIAM J. Sci. Comput. , 18 (1997), pp. 1040–1055]). Theoretical bounds, however, show that any realistic choice for α, in which both resolution and accuracy considerations are imposed, permits no more than a doubling of the time step fora stable explicit integrator in time, much less than the O ( N ) improvement claimed by Kosloff and Tal-Ezer. On the other hand, by choosing α carefully, it is possible to improve on the resolution of the Chebyshev method; in particular, one may achieve satisfactory resolution with fewer than π points per wavelength. Moreover, this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about 8 points per wavelength. Our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one-dimensional wave equations. Specifically, while α can be chosen such that the mapped method improves the accuracy and resolution of the Chebyshev method, for practical choices of N , it is not possible to achieve both single precision accuracy and gain the advantage of an O ( N −M ) time step.",

keywords = "Chebyshev collocation, accuracy, partial differential equations, stability",

author = "Jodi Mead and Renaut, {Rosemary A.}",

year = "2002",

month = jan,

day = "1",

language = "American English",

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TY - JOUR

T1 - Accuracy, Resolution and Stability Properties of a Modified Chebyshev Method

AU - Mead, Jodi

AU - Renaut, Rosemary A.

PY - 2002/1/1

Y1 - 2002/1/1

N2 - While the Chebyshev pseudospectral method provides a spectrally accurate method, integration of partial differential equations with spatial derivatives of order M requires time steps of approximately O(N −2M ) for stable explicit solvers. Theoretically, time steps may be increased to O ( N −M ) with the use of a parameter, α-dependent mapped method introduced by Kosloff and Tal-Ezer [ J. Comput. Phys. , 104 (1993), pp. 457–469]. Our analysis focuses on the utilization of this method for reasonable practical choices for N , namely N &lsim; 30, as may be needed for two- or three dimensional modeling. Results presented confirm that spectral accuracy with increasing N is possible both for constant α (Hesthaven, Dinesen, and Lynov [ J. Comput. Phys. , 155 (1999), pp. 287–306]) and for α scaled with N , α sufficiently different from 1 (Don and Solomonoff [ SIAM J. Sci. Comput. , 18 (1997), pp. 1040–1055]). Theoretical bounds, however, show that any realistic choice for α, in which both resolution and accuracy considerations are imposed, permits no more than a doubling of the time step fora stable explicit integrator in time, much less than the O ( N ) improvement claimed by Kosloff and Tal-Ezer. On the other hand, by choosing α carefully, it is possible to improve on the resolution of the Chebyshev method; in particular, one may achieve satisfactory resolution with fewer than π points per wavelength. Moreover, this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about 8 points per wavelength. Our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one-dimensional wave equations. Specifically, while α can be chosen such that the mapped method improves the accuracy and resolution of the Chebyshev method, for practical choices of N , it is not possible to achieve both single precision accuracy and gain the advantage of an O ( N −M ) time step.

AB - While the Chebyshev pseudospectral method provides a spectrally accurate method, integration of partial differential equations with spatial derivatives of order M requires time steps of approximately O(N −2M ) for stable explicit solvers. Theoretically, time steps may be increased to O ( N −M ) with the use of a parameter, α-dependent mapped method introduced by Kosloff and Tal-Ezer [ J. Comput. Phys. , 104 (1993), pp. 457–469]. Our analysis focuses on the utilization of this method for reasonable practical choices for N , namely N &lsim; 30, as may be needed for two- or three dimensional modeling. Results presented confirm that spectral accuracy with increasing N is possible both for constant α (Hesthaven, Dinesen, and Lynov [ J. Comput. Phys. , 155 (1999), pp. 287–306]) and for α scaled with N , α sufficiently different from 1 (Don and Solomonoff [ SIAM J. Sci. Comput. , 18 (1997), pp. 1040–1055]). Theoretical bounds, however, show that any realistic choice for α, in which both resolution and accuracy considerations are imposed, permits no more than a doubling of the time step fora stable explicit integrator in time, much less than the O ( N ) improvement claimed by Kosloff and Tal-Ezer. On the other hand, by choosing α carefully, it is possible to improve on the resolution of the Chebyshev method; in particular, one may achieve satisfactory resolution with fewer than π points per wavelength. Moreover, this improvement is noted not only for waves with the minimal resolution but also for waves sampled up to about 8 points per wavelength. Our conclusions are verified by calculation of phase and amplitude errors for numerical solutions of first and second order one-dimensional wave equations. Specifically, while α can be chosen such that the mapped method improves the accuracy and resolution of the Chebyshev method, for practical choices of N , it is not possible to achieve both single precision accuracy and gain the advantage of an O ( N −M ) time step.

KW - Chebyshev collocation

KW - accuracy

KW - partial differential equations

KW - stability

UR - https://scholarworks.boisestate.edu/math_facpubs/20

M3 - Article

SN - 1064-8275

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

ER -

Accuracy, Resolution and Stability Properties of a Modified Chebyshev Method

Abstract

Keywords

EGS Disciplines

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