Spectral and finite difference solutions of the Burgers equation

By C. Basdevant, M. Deville, P. Haldenwang, J. Lacroix, J. Quazzani. 1986

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Citations in the corpus

Neural-net-induced Gaussian process regression for function approximation and PDE solution (2018, methodology): “The blue solid curve is the exact solution computed according to [17], the red circle is the training points at n = 0 (namely N0 = 24), and the red dashed curve is the numerical solution.”
Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations (2018, background): “Let us start with the Burgers’ equation arising in various areas of engineering and applied mathematics, including ﬂuid mechanics, nonlinear acoustics, gas dynamics, and traﬃc ﬂow [36].”
Hidden physics models: Machine learning of nonlinear partial differential equations (2017, background): “Burgers’ equation arises in various areas of applied mathematics, including ﬂuid mechanics, nonlinear acoustics, gas dynamics, and traﬃc ﬂow [29].”
Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations (2017, background): “As a motivating example, the one dimensional Burgers’ equation [10] corresponds to the case where N [u;λ] = λ1uux − λ2uxx and λ = (λ1, λ2).”
Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations (2017, methodology): “This equation arises in various areas of applied mathematics, including fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow [14].”
Numerical Gaussian Processes for Time-Dependent and Nonlinear Partial Differential Equations (2017, background): “01/π—a value leading to the development of a nonsingular thin internal layer at x = 0 that is notoriously hard to resolve by classical numerical methods [4].”
Numerical solution of evolution equations by the Haar wavelet method (2007, methodology): “[1]), but it is quite difficult to compute.”
Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations (2021, methodology): “The exact solution is analytically available by the use of the Hopf-Cole transform (see [61] for details, which are omitted here for brevity).”
Message Passing Neural PDE Solvers (2022, unknown): “(53) Solutions are (Basdevant et al., 1986) u(t, x) = − ∫∞ −∞ sinπ(x− η)f(x− η) exp(−η 2/4νt)dη∫∞ −∞ f(x− η) exp(−η2/4νt)dη , (54) with f(y) = exp(− cos( πy2πν ) .”
Extended Physics-informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition based Deep Learning Framework for Nonlinear Partial Differential Equations (2020, background): “N (z) = WkΦ(ak−1N k−1(z))+bk ∈ Rk , 2 ≤ k ≤ L (2) andN (1)(z) = W(1)z + b(1), where in the last layer, the activation function is identity.”
Physics-informed deep generative models (2018, unknown): “This equation arises in various areas of applied mathematics, including fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow [29].”