A variable-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e6916" altimg="si281.svg"><mml:mi>θ</mml:mi></mml:math> method for parabolic problems of nonsmooth data

Lee, Philku; Kim, Seongjai

doi:10.1016/j.camwa.2019.08.006

Cited by 4 publications

(10 citation statements)

References 14 publications

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“…However, the CN method applied for nonsmooth data may introduce spurious oscillations to the numerical solution unless the algorithm parameters satisfy the maximum principle [22,23]. As analyzed by the authors [24], the undesired oscillations are due to instability involved in the explicit half step of the CN method, the first term in the right side of (13). e variable θ-method proposed in [24] suppresses spurious oscillations, by evolving the solution implicitly (θ ij � 1) at points x ij where the solution shows a certain portent of oscillations or reduced smoothness, and maintains as a similar accuracy as the CN method with smooth data.…”

Section: E θ-Method: Difference Equationmentioning

confidence: 99%

“…Figure 1 depicts the exact and numerical solutions evolved by the CN and the variable-θ method [24], while Figure 2 compares the numerical solutions at T � 1.0 obtained by the three numerical methods. e numerical solutions are obtained with Δt � 0.01 and Δx � Δy � 0.025.…”

Section: Numerical Oscillations Of the Cn Methodmentioning

confidence: 99%

“…e variable-θ method is analyzed for its numerical stability and accuracy and verified for various examples [24]. It results in nonoscillatory numerical solutions of which the accuracy is almost second-order in time.…”

Section: E Variable-θ Methodmentioning

confidence: 99%

“…We present here the main steps of variable-θ method for a nonoscillatory solution of (20); a complete study of the method for diffusion equation was published in [24].…”

Section: Linearization Through Extrapolationmentioning

confidence: 99%

“…e CN method and its perturbations (such as ADI) must be applied with care when the solution involves fast transitions or sharp edges; in particular, the time step size should be set very small, e.g., Δt � O(Δx 2 ), where Δx is the spatial grid size. In order to overcome the oscillation problem of the CN method applied for linear parabolic problems of nonsmooth data, the authors recently suggested a variable-θmethod in which the time-stepping parameter of the conventional θ-method, θ ∈ [0, 1], was determined based on local oscillatory characteristics of the solution and the data [24].…”

Section: Introductionmentioning

confidence: 99%

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A Nonoscillatory Second-Order Time-Stepping Procedure for Reaction-Diffusion Equations

2020

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After a theory of morphogenesis in chemical cells was introduced in the 1950s, much attention had been devoted to the numerical solution of reaction-diffusion (RD) partial differential equations (PDEs). The Crank–Nicolson (CN) method has been a common second-order time-stepping procedure. However, the CN method may introduce spurious oscillations for nonsmooth data unless the time step size is sufficiently small. This article studies a nonoscillatory second-order time-stepping procedure for RD equations, called a variable-θmethod, as a perturbation of the CN method. In each time level, the new method detects points of potential oscillations to implicitly resolve the solution there. The proposed time-stepping procedure is nonoscillatory and of a second-order temporal accuracy. Various examples are given to show effectiveness of the method. The article also performs a sensitivity analysis for the numerical solution of biological pattern forming models to conclude that the numerical solution is much more sensitive to the spatial mesh resolution than the temporal one. As the spatial resolution becomes higher for an improved accuracy, the CN method may produce spurious oscillations, while the proposed method results in stable solutions.

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Section: E θ-Method: Difference Equationmentioning

confidence: 99%

Section: Numerical Oscillations Of the Cn Methodmentioning

confidence: 99%

Section: E Variable-θ Methodmentioning

confidence: 99%

“…We present here the main steps of variable-θ method for a nonoscillatory solution of (20); a complete study of the method for diffusion equation was published in [24].…”

Section: Linearization Through Extrapolationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Nonoscillatory Second-Order Time-Stepping Procedure for Reaction-Diffusion Equations

2020

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Stable rotational symmetric schemes for nonlinear reaction-diffusion equations

Lee

Popescu

Kim

2022

Computers & Mathematics with Applications

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Variational quantum evolution equation solver

Leong

Ewe

Koh

2022

Sci Rep

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Variational quantum algorithms offer a promising new paradigm for solving partial differential equations on near-term quantum computers. Here, we propose a variational quantum algorithm for solving a general evolution equation through implicit time-stepping of the Laplacian operator. The use of encoded source states informed by preceding solution vectors results in faster convergence compared to random re-initialization. Through statevector simulations of the heat equation, we demonstrate how the time complexity of our algorithm scales with the Ansatz volume for gradient estimation and how the time-to-solution scales with the diffusion parameter. Our proposed algorithm extends economically to higher-order time-stepping schemes, such as the Crank–Nicolson method. We present a semi-implicit scheme for solving systems of evolution equations with non-linear terms, such as the reaction–diffusion and the incompressible Navier–Stokes equations, and demonstrate its validity by proof-of-concept results.

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A variable-θ method for parabolic problems of nonsmooth data

Cited by 4 publications

References 14 publications

A Nonoscillatory Second-Order Time-Stepping Procedure for Reaction-Diffusion Equations

A Nonoscillatory Second-Order Time-Stepping Procedure for Reaction-Diffusion Equations

Stable rotational symmetric schemes for nonlinear reaction-diffusion equations

Variational quantum evolution equation solver

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