ℒ-Splines and Viscosity Limits for Well-Balanced Schemes Acting on Linear Parabolic Equations

Gosse, Laurent

doi:10.1007/s10440-017-0122-5

Cited by 12 publications

(12 citation statements)

References 55 publications

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“…According to the definition of f , the following boundary value problem can be written to determine the numerical flux:

\begin{align} k_{1} \cdot \frac{\partial u}{\partial x} prefix- a_{1} \cdot u & = f_{j + \frac{1}{2}}, \forall x \in 𝒞_{j}^{⋆}, \forall j \in true{2, \dots, N prefix- 1 true}, \end{align}

\begin{align} u & = u_{j}, x = x_{j}, \end{align}

\begin{align} u & = u_{j + 1}, x = x_{j + 1} . \end{align}

It is important to remark that the boundary value problem (4) has two unknowns

f_{j + \frac{1}{2}}

and

u : 𝒞^{⋆} \overset{def}{:=} \cup_{j \in [[1, N]]} 𝒞_{j}^{⋆} \to R

with two boundary conditions. Therefore, the exact expression of the flux in the dual cell can be computed as:

f_{j + \frac{1}{2}} = \frac{d_{1}}{Δ x} \cdot [ℬ ()]

…”

Section: The Numerical Modelmentioning

confidence: 99%

“…The numerical stability condition of the system of Equation (3) is therefore given by the scheme used for Equations and . The SCHARFETTER‐GUMMEL scheme combined with the EULER explicit in time approach has the following stability restriction:

\begin{align} normalΔ t \underset{1 ⩽ k, l ⩽ 2}{normalmax} ((), \underset{1 ⩽ j ⩽ N}{normalmax} k_{k l, j} \underset{1 ⩽ j ⩽ N}{normalmax} ([], \frac{a_{k l, j}}{k_{k l, j}} normaltanh {((), \frac{a_{k l, j} normalΔ x}{2 k_{k l, j}})}^{prefix- 1})) ⩽ normalΔ x . \end{align}

…”

Section: The Numerical Modelmentioning

confidence: 99%

“…The numerical stability condition of the system of Equation (3) is therefore given by the scheme used for Equations (3a) and (3b). The SCHARFETTER-GUMMEL scheme combined with the EULER explicit in time approach has the following stability restriction: 29 Δt max…”

Section: Important Features Of the Numerical Schemementioning

confidence: 99%

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An efficient numerical model for the simulation of coupled heat, air, and moisture transfer in porous media

Berger

Dutykh

Mendes

et al. 2020

Engineering Reports

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This article proposes an efficient explicit numerical model with a relaxed stability condition for the simulation of heat, air, and moisture transfer in porous material. Three innovative approaches are combined to solve the system of two differential advection‐diffusion equations coupled with a purely diffusive equation. First, the DU FORT‐FRANKEL scheme is used to solve the diffusion equation, providing an explicit scheme with an extended stability region. Then, the two advection‐diffusion equations are solved using both the SCHARFETTER‐GUMMEL numerical scheme for the space discretisation and the two‐step RUNGE‐KUTTA method for the time variable. This combination enables to relax the stability condition by one order. The proposed numerical model is evaluated on three case studies. The first one considers quasi‐linear coefficients. The theoretical results of the numerical schemes are confirmed by our computations. Indeed, the stability condition is relaxed by a factor of 40 compared to the standard EULER explicit approach. The second case provides an analytical solution for a weakly nonlinear problem. A very satisfactory accuracy is observed between the reference solution and the one provided by the numerical model. The last case study assumes a more realistic application with nonlinear coefficients and ROBIN‐type boundary conditions. The computational time is reduced 10 times by using the proposed model in comparison with the explicit EULER method.

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“…According to the definition of f , the following boundary value problem can be written to determine the numerical flux:

\begin{align} k_{1} \cdot \frac{\partial u}{\partial x} prefix- a_{1} \cdot u & = f_{j + \frac{1}{2}}, \forall x \in 𝒞_{j}^{⋆}, \forall j \in true{2, \dots, N prefix- 1 true}, \end{align}

\begin{align} u & = u_{j}, x = x_{j}, \end{align}

\begin{align} u & = u_{j + 1}, x = x_{j + 1} . \end{align}

It is important to remark that the boundary value problem (4) has two unknowns

f_{j + \frac{1}{2}}

and

u : 𝒞^{⋆} \overset{def}{:=} \cup_{j \in [[1, N]]} 𝒞_{j}^{⋆} \to R

with two boundary conditions. Therefore, the exact expression of the flux in the dual cell can be computed as:

f_{j + \frac{1}{2}} = \frac{d_{1}}{Δ x} \cdot [ℬ ()]

…”

Section: The Numerical Modelmentioning

confidence: 99%

\begin{align} normalΔ t \underset{1 ⩽ k, l ⩽ 2}{normalmax} ((), \underset{1 ⩽ j ⩽ N}{normalmax} k_{k l, j} \underset{1 ⩽ j ⩽ N}{normalmax} ([], \frac{a_{k l, j}}{k_{k l, j}} normaltanh {((), \frac{a_{k l, j} normalΔ x}{2 k_{k l, j}})}^{prefix- 1})) ⩽ normalΔ x . \end{align}

…”

Section: The Numerical Modelmentioning

confidence: 99%

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An efficient numerical model for the simulation of coupled heat, air, and moisture transfer in porous media

Berger

Dutykh

Mendes

et al. 2020

Engineering Reports

Self Cite

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show abstract

“…1.1. Finite-differences based on L-splines and borrowed from [18] were set up in order to maximize both stability and accuracy in this context of a square-well potential: namely, we implemented both, a = 0.35, b = 0.65 on the left, a = 0.35, b = 0.675 on the right, with ε = 0.001, 2 7 grid points, and the following discretization,…”

Section: 1mentioning

confidence: 99%

“…meaning that the solution to (4.2) is completely known. • At this point, the main idea to derive the Steklov scheme, see [17,18], is based on the discretization of the normal derivative of the solution v(r, θ) in…”

Section: Derivation Of the Numerical Processmentioning

confidence: 99%

A Two-Dimensional “Flea on the Elephant” Phenomenon and its Numerical Visualization

Bianchini¹,

Gosse²,

Zuazua³

2019

Multiscale Model. Simul.

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Localization phenomena (sometimes called "flea on the elephant") for the operator L ε = −ε 2 ∆u + p(x)u, p(x) being an asymmetric double-well potential, are studied both analytically and numerically, mostly in two space dimensions within a perturbative framework. Starting from a classical harmonic potential, the effects of various perturbations are retrieved, especially in the case of two asymmetric potential wells. These findings are illustrated numerically by means of an original algorithm, which relies on a discrete approximation of the Steklov-Poincaré operator for L ε , and for which error estimates are established. Such a two-dimensional discretization produces less mesh-imprinting than more standard finite-differences and captures correctly sharp layers.

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B-Spline Approximation for Polynomial Splines

Singh

Zaynidinov

2018

Signal Processing Applications Using Multidimensional Polynomial Splines

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The traditional tabular-polynomial methods for the approximation of complex functional relationships have a limitation on accuracy, smoothness, and they do not allow parallelisation of the computing structures for the implementation of approximation using classical polynomials on Horner's algorithm [1][2][3]. For instance, a second-level polynomial on Horner's algorithm can be represented in the following way:The structure, shown in Fig. 2.1, was proposed for the implementation of the second-level polynomials. The use of parabolic basic B-splines for the approximation of functions allows enhancing the smoothness and locality properties [4,5]. According to Formula (2.1.1), the value of the function under interpolation on a random point of a given interval is determined by the values m + 1 of summands (where m is the B-spline level)-pair derivatives of basic functions to constant coefficients.The local formulae are convenient among the coefficient determination methods in data processing with B-splines for they allow avoiding solution of equation systems [6]. At calculation of b coefficients using the three-point formula, only three values f i − 1 , f i , f i + 1 of the function are used. This facilitates the initiation of processing without waiting for the arrival of all data [7], i.e. a combination of data entry and processing operations.

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ℒ-Splines and Viscosity Limits for Well-Balanced Schemes Acting on Linear Parabolic Equations

Cited by 12 publications

References 55 publications

An efficient numerical model for the simulation of coupled heat, air, and moisture transfer in porous media

An efficient numerical model for the simulation of coupled heat, air, and moisture transfer in porous media

A Two-Dimensional “Flea on the Elephant” Phenomenon and its Numerical Visualization

B-Spline Approximation for Polynomial Splines

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