A high-order discontinuous Galerkin method for unsteady advection–diffusion problems

Borker, Raunak; Farhat, Charbel; Tezaur, Radek

doi:10.1016/j.jcp.2016.12.021

Cited by 13 publications

(10 citation statements)

References 25 publications

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“…The reason for this is that the solution develops very large gradients near the x = 1 boundary, and a sufficiently large DNN is needed to accurately approximate it. We note that in numerical discretization-based solutions of ADEs, an increase in P e often requires a finer mesh to maintain the same accuracy [38,51]. demonstrates that the final loss value increases with increasing P e, which is mainly due to the larger final value of the residual loss (L f ).…”

Section: One-dimensional Time-dependent Adementioning

confidence: 85%

“…First, we compare the PINN solutions of the 1D (Section 3.1) and 2D (Section 3.2) time-dependent ADEs with the analytical solutions that are commonly used to benchmark numerical grid-based methods [49,50,51,52]. In Section 3.3, we investigate the effect of grid orientation on the ADE solution with P e 1, where the crosswind diffusion could lead to numerical instabilities in discretization-based methods.…”

Section: Forward Flow and Advection-dispersion Equationsmentioning

confidence: 99%

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Physics-Informed Neural Network Method for Forward and Backward Advection-Dispersion Equations

He,

Tartakovsky

2020

Preprint

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We propose a discretization-free approach based on the physics-informed neural network (PINN) method for solving coupled advection-dispersion and Darcy flow equations with space-dependent hydraulic conductivity. In this approach, the hydraulic conductivity, hydraulic head, and concentration fields are approximated with deep neural networks (DNNs). We assume that the conductivity field is given by its values on a grid, and we use these values to train the conductivity DNN. The head and concentration DNNs are trained by minimizing the residuals of the flow equation and ADE and using the initial and boundary conditions as additional constraints. The PINN method is applied to one-and two-dimensional forward advection dispersion equations (ADEs), where its performance for various Péclet numbers (P e) is compared with the analytical and numerical solutions.We find that the PINN method is accurate with errors of less than 1% and outperforms some conventional discretization-based methods for P e larger than 100. Next, we demonstrate that the PINN method remains accurate for the backward ADEs, with the relative errors in most cases staying under 5% compared to the reference concentration field. Finally, we show that when available, the concentration measurements can be easily incorporated in the PINN method and significantly improve (by more than 50% in the considered cases) the accuracy

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Section: One-dimensional Time-dependent Adementioning

confidence: 85%

Section: Forward Flow and Advection-dispersion Equationsmentioning

confidence: 99%

Physics-Informed Neural Network Method for Forward and Backward Advection-Dispersion Equations

He,

Tartakovsky

2020

Preprint

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“…The limitations of the PIELM will be further investigated in the next section. We close this section by summarizing the advantages of the PIELM which are as follows: A potential reason for the failure of PIELM in solving advection-diffusion equation [23] could be the limited representation capacity of PIELM to represent a complex function. A PDE consists of function and its derivatives.…”

Section: The Non-physical Oscillations Don't Grow With Timementioning

confidence: 99%

“…However, PIELM doesn't impose any such restriction and we can take larger time steps.4.3.1 1D unsteady advection-diffusion [ TC-9 ]Exact and PIELM solution for unsteady 1D convection diffusion at ν = 0.005. Red: PIELM, Blue: Exact.The 1D equivalent of the unsteady 2D advection-diffusion equation solved by Borker et al[23] is given by…”

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confidence: 99%

“…PIELM solution for 2D unsteady advection diffusion Error for 2D unsteady advection diffusionWe solve the following unsteady 2D advection-diffusion equation[23] ut + au x + bu y = ν(u xx + u yy ), (x, y, t) Ω xy x[0, 0.2],(59)u(x, y, 0) = F (x, y), (x, y) Ω xy ,(60)u(x, y, t) = B(x, y, t), (x, y, t) ∂Ω xy x[0, 0.5], (61) where Ω xy = [0, 1]x[0, 1]. a = cos(22.5) and b = sin(22.5) are advection coefficients in x and y directions and ν = 0.005 is the diffusion coefficient.…”

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confidence: 99%

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Physics Informed Extreme Learning Machine (PIELM)–A rapid method for the numerical solution of partial differential equations

2020

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There has been rapid progress recently on the application of deep networks to solution of partial differential equations, collectively labelled as Physics Informed Neural Networks (PINNs). In this paper, we develop Physics Informed Extreme Learning Machine (PIELM), a rapid version of PINNs which can be applied to stationary and time dependent linear partial differential equations. We demonstrate that PIELM matches or exceeds the accuracy of PINNs on a range of problems. We also discuss the limitations of neural network based approaches, including our PIELM, in the solution of PDEs on large domains and suggest an extension, a distributed version of our algorithm --DPIELM. We show that DPIELM produces excellent results comparable to conventional numerical techniques in the solution of time-dependent problems. Collectively, this work contributes towards making the use of neural networks in the solution of partial differential equations in complex domains as a competitive alternative to conventional discretization techniques.Keywords Partial differential equations · Physics informed neural networks · Extreme learning machine · Advection-diffusion equation * Use footnote for providing further information about author (webpage, alternative address)-not for acknowledging funding agencies.

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A finite element formulation preserving symmetric and banded diffusion stiffness matrix characteristics for fractional differential equations

Lin

Wang

2017

Comput Mech

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A high-order discontinuous Galerkin method for unsteady advection–diffusion problems

Cited by 13 publications

References 25 publications

Physics-Informed Neural Network Method for Forward and Backward Advection-Dispersion Equations

Physics-Informed Neural Network Method for Forward and Backward Advection-Dispersion Equations

Physics Informed Extreme Learning Machine (PIELM)–A rapid method for the numerical solution of partial differential equations

A finite element formulation preserving symmetric and banded diffusion stiffness matrix characteristics for fractional differential equations

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