Symmetries of the discrete Burgers equation

We propose an integrable discrete model of one‐dimensional soil water infiltration. This model is based on the continuum model by Broadbridge and White, which takes the form of nonlinear convection–diffusion equation with a nonlinear flux boundary condition at the surface. It is transformed to the Burgers equation with a time‐dependent flux term by the hodograph transformation. We construct a discrete model preserving the underlying integrability, which is formulated as the self‐adaptive moving mesh scheme. The discretization is based on linearizability of the Burgers equation to the linear diffusion equation, but the naïve discretization based on the Euler scheme which is often used in the theory of discrete integrable systems does not necessarily give a good numerical scheme. Taking desirable properties of a numerical scheme into account, we propose an alternative discrete model that produces solutions with similar accuracy to direct computation on the original nonlinear equation, but with clear benefits regarding computational cost.

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“…To this end, it is convenient to write down the scheme in terms of

u_{n}^{m}

. We then have the discrete Burgers equation

\begin{matrix} true \\ \frac{u_{n}^{m + 1}}{u_{n}^{m}} = \frac{1 + \frac{δ}{ε^{2}} [u_{n + 1}^{m} - 2 + \frac{1}{u_{n}^{m}} + κ^{m} (() u_{n + 1}^{m} - \frac{1}{u_{n}^{m}})]}{1 + \frac{δ}{ε^{2}} [u_{n}^{m} - 2 + \frac{1}{u_{n - 1}^{m}} + κ^{m} (() u_{n}^{m} - \frac{1}{u_{n - 1}^{m}})]}, \\ κ^{m} = \frac{ε (R^{m} - β)}{2 a^{\frac{1}{2}}}, \\ n = 1, \dots, N - 1, m = 0, 1, 2, \dots, \end{matrix}

with initial condition

u_{n}^{0} = \frac{1 - \frac{2 a^{\frac{1}{2}}}{b ε} μ_{n}^{(0)}}{1 + \frac{2 a^{\frac{1}{2}}}{b ε} μ_{n}^{(0)}}, n = 0, 2 \dots, N - 1,

and the boundary conditions

u_{0}^{m} = \frac{1 - κ^{m}}{2 - a ε^{2} - false(1 + κ^{...}}

…”

Section: Integrable Discrete Modelsmentioning

confidence: 99%

Integrable Discrete Model for One‐Dimensional Soil Water Infiltration

et al. 2018

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“…Similar techniques can be applied to the previously introduced Burgers' equation (1). In Heredero, Levi, and Winternitz (1999) it is shown that the standard Burgers' equation with = 1,…”

Section: Parameter Function = (X)mentioning

confidence: 99%

A heat equation for freezing processes with phase change: stability analysis and applications

Backi

Bendtsen

Leth

et al. 2015

International Journal of Control

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In this work the stability properties as well as possible applications of a partial differential equation (PDE) with state-dependent parameters are investigated. Among other things, the PDE describes freezing of foodstuff, and is closely related to the (Potential) Burgers' Equation. We show that for certain forms of coefficient functions, the PDE converges to a stationary solution given by (fixed) boundary conditions that make physical sense. These boundary conditions are either symmetric or asymmetric of Dirichlet type. Furthermore we present an observer design based on the PDE model for estimation of innerdomain temperatures in block-frozen fish and for monitoring freezing time. We illustrate the results with numerical simulations.

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“…Heredero et al (1999) showed that the standard Burgers' equation with parameters set to 1, u t = u xx + 2uu x (4) can be transformed into the Potential Burgers' equation by using the transformation u = v x , resulting in v tx = v xxx + 2v x v xx . After integrating this expression we obtain…”

Section: Preliminariesmentioning

confidence: 99%

The nonlinear heat equation with state-dependent parameters and its connection to the Burgers' and the potential Burgers' equation

Backi

Bendtsen

Leth

et al. 2014

IFAC Proceedings Volumes

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In this work the stability properties of a nonlinear partial differential equation (PDE) with state-dependent parameters is investigated. Among other things, the PDE describes freezing of foodstuff, and is closely related to the (Potential) Burgers' Equation. We show that for certain forms of coefficient functions, the PDE converges to a stationary solution given by (fixed) boundary conditions that make physical sense. We illustrate the results with numerical simulations.

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Symmetries of the discrete Burgers equation

Cited by 56 publications

References 20 publications

Integrable Discrete Model for One‐Dimensional Soil Water Infiltration

Integrable Discrete Model for One‐Dimensional Soil Water Infiltration

A heat equation for freezing processes with phase change: stability analysis and applications

The nonlinear heat equation with state-dependent parameters and its connection to the Burgers' and the potential Burgers' equation

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