An Error Corrected Euler Method for Solving Stiff Problems Based on Chebyshev Collocation

Kim, Philsu; Piao, Xiangfan; Kim, Sang Dong

doi:10.1137/100808691

Cited by 27 publications

(30 citation statements)

References 30 publications

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“…Hence, we are now in a position to apply ECEM developed in [22] for time discretizations. For this purpose, consider a general nonlinear stiff system of ODEs instead of (2.4):…”

Section: For a Timementioning

confidence: 99%

“…Recalling that the Euler method has the local truncation error O(τ 2 ), one can guess the deleted error term in (3.6) may be quite small and can be ignored. In fact, one may prove that the error is O(τ 5 ) provided λ = O(τ 2 ) (see [22]). It is remarkable that once the error term is disregarded, the system (3.6) will be completely linear and its approximation scheme becomes an explicit type.…”

Section: ξ(T) Is a Function Between Y(t) And φ(T) ∇F(t Y(t)) And ∇ mentioning

confidence: 99%

“…For an approximation of (3.6), the Chebyshev-collocation method (CCM) will be used because it is known that CCM have a good stability for the stiff system (2.5) (see [22,29]). Let…”

Section: ξ(T) Is a Function Between Y(t) And φ(T) ∇F(t Y(t)) And ∇ mentioning

confidence: 99%

“…whose error has the asymptotic behavior O(τ 5 ) if ψ i (t) is fifth-times continuously differential function [22]. By substituting (3.8) into (3.6), one may approximate the equations (3.6) with (3.9) …”

Section: ξ(T) Is a Function Between Y(t) And φ(T) ∇F(t Y(t)) And ∇ mentioning

confidence: 99%

“…Hence the primary goal of the present paper is to construct a non-standard type of an explicit method providing a nice spatial discretization, which does not ask to solve any extra nonlinear equations required by an implicit method. The error corrected Euler method (ECEM) with convergence order 4 for the scalar stiff initial value problem developed by authors [22] is extended to solve a system of ODEs in this paper. To confirm the effectiveness and convergence, the present method is compared with several known methods.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A New Time Stepping Method for Solving One Dimensional Burgers' Equations

Piao¹,

Kim²,

Kim³

et al. 2012

Kyungpook mathematical journal

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Abstract. In this paper, we present a simple explicit type numerical method for discretizations in time for solving one dimensional Burgers' equations. The proposed method does not need an iteration process that may be required in most implicit methods and have good convergence and efficiency in computational sense compared to other known numerical methods. For evidences, several numerical demonstrations are also provided.

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“…Hence, we are now in a position to apply ECEM developed in [22] for time discretizations. For this purpose, consider a general nonlinear stiff system of ODEs instead of (2.4):…”

Section: For a Timementioning

confidence: 99%

Section: ξ(T) Is a Function Between Y(t) And φ(T) ∇F(t Y(t)) And ∇ mentioning

confidence: 99%

“…For an approximation of (3.6), the Chebyshev-collocation method (CCM) will be used because it is known that CCM have a good stability for the stiff system (2.5) (see [22,29]). Let…”

Section: ξ(T) Is a Function Between Y(t) And φ(T) ∇F(t Y(t)) And ∇ mentioning

confidence: 99%

Section: ξ(T) Is a Function Between Y(t) And φ(T) ∇F(t Y(t)) And ∇ mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A New Time Stepping Method for Solving One Dimensional Burgers' Equations

Piao¹,

Kim²,

Kim³

et al. 2012

Kyungpook mathematical journal

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High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

Bak

2019

Numerical Methods Partial

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In this study, new high‐order backward semi‐Lagrangian methods are developed to solve nonlinear advection–diffusion type problems, which are realized using high‐order characteristic‐tracking strategies. The proposed characteristic‐tracking strategies are second‐order L‐stable and third‐order L(α)‐stable methods, which are based on a classical implicit multistep method combined with a error‐correction method. We also use backward differentiation formulas and the fourth‐order finite‐difference scheme for diffusion problem discretization in the temporal and spatial domains, respectively. To demonstrate the adaptability and efficiency of these time‐discretization strategies, we apply these methods to nonlinear advection–diffusion type problems such as the viscous Burgers' equation. Through simulations, not only the temporal and spatial accuracies are numerically evaluated but also the proposed methods are shown to be superior to the compared existing characteristic‐tracking methods under the same rates of convergence in terms of accuracy and efficiency. Finally, we have shown that the proposed method well preserves the energy and mass when the viscosity coefficient becomes zero.

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Numerical Methods

Kuehn

2014

Applied Mathematical Sciences

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An Error Corrected Euler Method for Solving Stiff Problems Based on Chebyshev Collocation

Cited by 27 publications

References 30 publications

A New Time Stepping Method for Solving One Dimensional Burgers' Equations

A New Time Stepping Method for Solving One Dimensional Burgers' Equations

High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

Numerical Methods

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