Viscous Burgers Research Articles

Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation

<p style='text-indent:20px;'>We consider one-dimensional distributed optimal control problems with the state equation being given by the viscous Burgers equation. We discretize using a space-time discontinuous Galerkin approach. We use upwind flux in time and the symmetric interior penalty approach for discretizing the viscous term. Our focus is on the discretization of the convection terms. We aim for using conservative discretizations for the convection terms in both the state and the adjoint equation, while ensuring that the approaches of discretize-then-optimize and optimize-then-discretize commute. We show that this is possible if the arising source term in the adjoint equation is discretized properly, following the ideas of well-balanced discretizations for balance laws. We support our findings by numerical results.</p>

Stabilized neural ordinary differential equations for long-time forecasting of dynamical systems

In data-driven modeling of spatiotemporal phenomena careful consideration is needed in capturing the dynamics of the high wavenumbers. This problem becomes especially challenging when the system of interest exhibits shocks or chaotic dynamics. We present a data-driven modeling method that accurately captures shocks and chaotic dynamics by proposing a new architecture, stabilized neural ordinary differential equation (ODE). In our proposed architecture, we learn the right-hand-side (RHS) of an ODE by adding the outputs of two NN together where one learns a linear term and the other a nonlinear term. Specifically, we implement this by training a sparse linear convolutional NN to learn the linear term and a dense fully-connected nonlinear NN to learn the nonlinear term. This contrasts with the standard neural ODE which involves training a single NN for the RHS. We apply this setup to the viscous Burgers equation, which exhibits shocked behavior, and show stabilized neural ODEs provide better short-time tracking, prediction of the energy spectrum, and robustness to noisy initial conditions than standard neural ODEs. We also apply this method to chaotic trajectories of the Kuramoto-Sivashinsky equation. In this case, stabilized neural ODEs keep long-time trajectories on the attractor, and are highly robust to noisy initial conditions, while standard neural ODEs fail at achieving either of these results. We conclude by demonstrating how stabilizing neural ODEs provide a natural extension for use in reduced-order modeling by projecting the dynamics onto the eigenvectors of the learned linear term.

A hybrid physics-informed neural network for nonlinear partial differential equation

The recently developed physics-informed machine learning has made great progress for solving nonlinear partial differential equations (PDEs), however, it may fail to provide reasonable approximations to the PDEs with discontinuous solutions. In this paper, we focus on the discrete time physics-informed neural network (PINN), and propose a hybrid PINN (hPINN) scheme for the nonlinear PDEs. In this approach, the local solution structures are classified as smooth and nonsmooth scales by introducing a discontinuity indicator, and then the automatic differentiation technique is employed for resolving smooth scales, while an improved weighted essentially nonoscillatory (WENO) scheme is adopted to capture discontinuities. We then test the present approach by considering the viscous and inviscid Burgers equations, and it is shown that compared with original discrete time PINN, the present hPINN approach has a better performance in approximating the discontinuous solution even at a relatively larger time step.

Pointwise error analysis of the BDF3 compact finite difference scheme for viscous Burgers' equations

This paper formulates a third-order backward differentiation formula (BDF3) fourth-order compact difference scheme based on a developed fourth-order operator for computing the approximate solution of the Burger's equation. The equation is one of the useful description for modeling nonlinear acoustics, gas dynamics, fluid mechanics and etc. This proposed approach approximates the solution of Burger's equation with the help of two main steps. In the first step, the temporal discretization is accomplished by virtue of the BDF3 approach. In the second step, a developed fourth-order operator and the classic compact difference formula combine with the method of order reduction are applied for spatial discretization, thereby constructing a fully-discrete scheme. The theoretical analysis is proved in detail by means of the discrete energy method. The proposed scheme is convergent with third order for time and fourth order for space. Numerical results are carried out to verify the validity and accuracy of the proposed method.

The Analytical Solution of an Unsteady State Heat Transfer Model for the Confined Aquifer under the Influence of Water Temperature Variation in the River Channel

The effect of water temperature variation in a river channel on groundwater temperature in the confined aquifer it cuts can be generalized to a one-dimensional thermal convection-conduction problem in which the boundary water temperature rises instantaneously and then remains constant. The basic equation of thermal transport for such a problem is the viscous Burgers equation, which is difficult to solve analytically. To solve this problem, the Cole–Hopf transform was used to convert the second-order nonlinear thermal convection-conduction equation into a heat conduction equation with exponential function-type boundary conditions. Considering the difficulty of calculating the inverse of the image function of the boundary function, the characteristics and properties of the Laplace transform were used to derive the theoretical solution of the model without relying on the transformation of the boundary function, and the analytical solution was obtained by substituting the boundary condition into the theoretical solution. The analytical solution was used to analyze the temperature response laws of aquifers to parameter variation. Subsequently, a 40-day numerical simulation was conducted to analyze the boundary influence range and the results from the analytical method were compared to those from the numerical method. The study shows that: (1) the greater the distance from the river canal and the lower the aquifer flow velocity, the slower the aquifer temperature changes; (2) the influence range of the river canal boundary increases from 18.19 m to 23.19 m at the end of simulation period as the groundwater seepage velocity v increases from 0.08 m/d to 0.12 m/d; (3) the relative errors of the analytical and numerical methods are mostly less than 5%, confirming the rationality of the analytical solution.

A Bernstein–Bézier Lagrange–Galerkin method for three-dimensional advection-dominated problems

We present a high-order Bernstein–Bézier finite element discretization to accurately solve three-dimensional advection-dominated problems on unstructured tetrahedral meshes. The key idea consists of implementing a modified method of characteristics to discretize the advection terms in a Bernstein–Bésier finite element framework. The proposed Bernstein–Bézier Lagrange–Galerkin method has been designed so that the Courant–Friedrichs–Lewy condition is strongly relaxed using semi-Lagrangian time discretization. A low complexity procedures in building finite element matrices and load vectors is also achieved in the present work by both the analytical rule and the sum factorization method using the tensorial feature of Bernstein polynomials. Several numerical examples including advection–diffusion equations with known analytical solutions and the viscous Burgers problem are considered to illustrate the accuracy, robustness and performance of the proposed approach. The computed results support our expectations for a stable and highly accurate Bernstein–Bézier Lagrange–Galerkin finite element method for three-dimensional advection-dominated problems.

Global Thin Plate Spline Differential Quadrature as a Meshless Numerical Solution for Two Dimensional Viscous Burgers’ Equation

Global Thin Plate Spline Differential Quadrature as a Meshless Numerical Solution for Two Dimensional Viscous Burgers’ Equation

Variational multiscale reinforcement learning for discovering reduced order closure models of nonlinear spatiotemporal transport systems

A central challenge in the computational modeling and simulation of a multitude of science applications is to achieve robust and accurate closures for their coarse-grained representations due to underlying highly nonlinear multiscale interactions. These closure models are common in many nonlinear spatiotemporal systems to account for losses due to reduced order representations, including many transport phenomena in fluids. Previous data-driven closure modeling efforts have mostly focused on supervised learning approaches using high fidelity simulation data. On the other hand, reinforcement learning (RL) is a powerful yet relatively uncharted method in spatiotemporally extended systems. In this study, we put forth a modular dynamic closure modeling and discovery framework to stabilize the Galerkin projection based reduced order models that may arise in many nonlinear spatiotemporal dynamical systems with quadratic nonlinearity. However, a key element in creating a robust RL agent is to introduce a feasible reward function, which can be constituted of any difference metrics between the RL model and high fidelity simulation data. First, we introduce a multi-modal RL to discover mode-dependant closure policies that utilize the high fidelity data in rewarding our RL agent. We then formulate a variational multiscale RL (VMRL) approach to discover closure models without requiring access to the high fidelity data in designing the reward function. Specifically, our chief innovation is to leverage variational multiscale formalism to quantify the difference between modal interactions in Galerkin systems. Our results in simulating the viscous Burgers equation indicate that the proposed VMRL method leads to robust and accurate closure parameterizations, and it may potentially be used to discover scale-aware closure models for complex dynamical systems.

The totally volume local discontinuous Galerkin method for viscous Burgers’ equations.

In this paper, the totally volume local discontinuous Galerkin TV-LDG method is proposed to solve the viscous Burgers’ equations with appropriate initial and boundary conditions. Time derivative is discretized by the third-order strong stability preserving Runge Kutta explicit SSP-RK (3, 3) method. Space derivatives discretization is performed by the totally volume local discontinuous Galerkin method. Finally, the validity of proposed scheme is demonstrated by numerical experiments and shows that the present scheme offers better accuracy in comparison with other existing numerical methods.

A Finite Difference Approximation for Numerical Simulation of 2D Viscous Coupled Burgers Equations

Many of the physical phenomena in nature are usually expressed in terms of algebraic, differential or integral equations.Several nonlinear phenomena playing a very important role in engineering sciences, physics and computational mathematics are usually modeled by those non-linear partial differential equations (PDEs). It is usually difficult and problematic to examine and find out nalytical solutions of initial-boundary value problems consisting of PDEs. In fact, there is no a certain method or technique working well for all these type equations. For this reason, their approximate solutions are usually preferred rather than analytical ones of such type equations. Thus, many researchers are concentrated on approximate methods and techniques to obtain numerical solutions of non-linear PDEs. In the present article, the numerical simulation of the two-dimensional coupled Burgers equation (2D-cBE) has been sought by finite difference method based on Crank-Nicolson type approximation. Widely used three test examples given with appropriate initial and boundary conditions are used for the simulation process. During the simulation process,the error norms $L_{2}$, $L_{\infty}$ are calculated if the exact solutions are already known, otherwise the pointwise values and graphics are provided for comparison. The newly obtained error norms $L_{2}$, $L_{\infty}$ by the presented schemes are compared with those of some of the numerical solutions in the literature. A good consistency and accuracy are observed both by numerical values and visual illustrations.

Fredholm transformation on Laplacian and rapid stabilization for the heat equation

We study the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. This classical framework allows us to present the backstepping method with Fredholm transformations for the Laplace operator in a sharp functional setting, which is the main objective of this work. We first prove that, under some assumptions on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. Then, we prove that the Fredholm transformation constructed for the Laplacian also leads to the local rapid stability of the viscous Burgers equation.

Semi-analytic shooting methods for Burgers’ equation

We implement new semi-analytic shooting methods for the stationary viscous Burgers’ equation by modifying the classical time differencing methods. When the viscosity is small, a very stiff boundary layer appears and this boundary layer causes significant difficulties to approximate the solution for Burgers’ equation. To overcome this issue and improve the numerical quality of the shooting methods with the classical Integrating Factor (IF) methods and Exponential Time Differencing (ETD) methods, we first employ the singular perturbation analysis for Burgers’ equation, and derived the so-called correctors that approximate the stiff part of the solution. Then, we build our new semi-analytic shooting methods for the stationary viscous Burgers’ equation by embedding these correctors into the IF and ETD methods. By performing numerical simulations, we verify that our new schemes, enriched with the correctors, give much better approximations, compared with the classical schemes.

Lack of controllability of the viscous Burgers equation: part I—the $$\mathrm{L}^\infty $$ setting

Lack of controllability of the viscous Burgers equation: part I—the $$\mathrm{L}^\infty $$ setting

A Model-Constrained Tangent Slope Learning Approach for Dynamical Systems

Real-time accurate solutions of large-scale complex dynamical systems are in critical need for control, optimisation, uncertainty quantification, and decision-making in practical engineering and science applications, especially digital twin applications. This paper contributes in this direction a model-constrained tangent slope learning (mcTangent) approach. At the heart of mcTangent is the synergy of several desirable strategies: (i) a tangent slope learning to take advantage of the neural network speed and the time-accurate nature of the method of lines; (ii) a model-constrained approach to encode the neural network tangent slope with the underlying governing equations; (iii) sequential learning strategies to promote long-time stability and accuracy; and (iv) data randomisation approach to implicitly enforce the smoothness of the neural network tangent slope and its likeliness to the truth tangent slope up second order derivatives in order to further enhance the stability and accuracy of mcTangent solutions. Rigorous results are provided to analyse and justify the proposed approach. Several numerical results for transport equation, viscous Burgers equation, and Navier–Stokes equation are presented to study and demonstrate the robustness and long-time accuracy of the proposed mcTangent learning approach.

Bernstein-Bézier Galerkin-Characteristics Finite Element Method for Convection-Diffusion Problems

A class of Bernstein-Bézier basis based high-order finite element methods is developed for the Galerkin-characteristics solution of convection-diffusion problems. The Galerkin-characteristics formulation is derived using a semi-Lagrangian discretization of the total derivative in the considered problems. The spatial discretization is performed using the finite element method on unstructured meshes. The Lagrangian interpretation in this approach greatly reduces the time truncation errors in the Eulerian methods. To achieve high-order accuracy in the Galerkin-characteristics solver, the semi-Lagrangian method requires high-order interpolating procedures. In the present work, this step is carried out using the Bernstein-Bézier basis functions to evaluate the solution at the departure points. Triangular Bernstein-Bézier patches are constructed in a simple and inherent manner over finite elements along the characteristics. An efficient preconditioned conjugate gradient solver is used for the linear systems of algebraic equations. Several numerical examples including advection-diffusion equations with known analytical solutions and the viscous Burgers problem are considered to illustrate the accuracy, robustness and performance of the proposed approach. The computed results support our expectations for a stable and highly accurate Bernstein-Bézier Galerkin-characteristics finite element method for convection-diffusion problems.

Existence theorem for global in time solutions to Burgers equation with a time delay

In this paper, we consider the one-dimensional Cauchy problem of the generalized viscous Burgers equations with a time delay and analyze a delay effect. We prove the existence of the global in time solution when the product of the size of the delay parameter and the one of the initial history is suitably small. Moreover we show theorem concerning with the regularity of the global in time solution.

Predicting waves in fluids with deep neural network

In this paper, we present a deep learning technique for data-driven predictions of wave propagation in a fluid medium. The technique relies on an attention-based convolutional recurrent autoencoder network (AB-CRAN). To construct a low-dimensional representation of wave propagation data, we employ a denoising-based convolutional autoencoder. The AB-CRAN architecture with attention-based long short-term memory cells forms our deep neural network model for the time marching of the low-dimensional features. We assess the proposed AB-CRAN framework against the standard recurrent neural network for the low-dimensional learning of wave propagation. To demonstrate the effectiveness of the AB-CRAN model, we consider three benchmark problems, namely, one-dimensional linear convection, the nonlinear viscous Burgers equation, and the two-dimensional Saint-Venant shallow water system. Using the spatial-temporal datasets from the benchmark problems, our novel AB-CRAN architecture accurately captures the wave amplitude and preserves the wave characteristics of the solution for long time horizons. The attention-based sequence-to-sequence network increases the time-horizon of prediction compared to the standard recurrent neural network with long short-term memory cells. The denoising autoencoder further reduces the mean squared error of prediction and improves the generalization capability in the parameter space.

Numerical solution of Coupled Viscous Burgers’ equations using RBF-QR method

Numerical solution of Coupled Viscous Burgers’ equations using RBF-QR method

An economical robust algorithm for solving 1D coupled Burgers’ equations in a semi-Lagrangian framework

In this study, we propose an efficient algorithm for solving one-dimensional coupled viscous Burgers’ equations. One of the main accomplishments of this study is to develop a stable high-order algorithm for the system of reaction–diffusion equations. The algorithm is “robust” because it is designed to prevent non-physical oscillations through an iteration procedure of a block Gauss-Seidel type. The other is to develop an efficient algorithm for the Cauchy problem. For this, we first find half of the upstream points by adopting a multi-step qth-order (q=2,3) error correction method. The algorithm is also “economical” in the sense that an interpolation strategy for finding the remaining upstream points is designed to dramatically reduce the high computational cost for solving the nonlinear Cauchy problem without damage to the order of accuracy. Three benchmark problems are simulated to investigate the accuracy and the superiority of the proposed method. It turns out that the proposed method numerically has the qth-order temporal and 4th-order spatial accuracies. In addition, the numerical experiments show that the proposed method is superior to the compared methods in the sense of the computational cost.

Shock trace prediction by reduced models for a viscous stochastic Burgers equation.

Viscous shocks are a particular type of extreme event in nonlinear multiscale systems, and their representation requires small scales. Model reduction can thus play an essential role in reducing the computational cost for the prediction of shocks. Yet, reduced models typically aim to approximate large-scale dominating dynamics, which do not resolve the small scales by design. To resolve this representation barrier, we introduce a new qualitative characterization of the space-time locations of shocks, named the "shock trace," via a space-time indicator function based on an empirical resolution-adaptive threshold. Unlike exact shocks, the shock traces can be captured within the representation capacity of the large scales, thus facilitating the forecast of the timing and locations of the shocks utilizing reduced models. Within the context of a viscous stochastic Burgers equation, we show that a data-driven reduced model, in the form of nonlinear autoregression (NAR) time series models, can accurately predict the random shock traces, with relatively low rates of false predictions. Furthermore, the NAR model, which includes nonlinear closure terms to approximate the feedback from the small scales, significantly outperforms the corresponding Galerkin truncated model in the scenario of either noiseless or noisy observations. The results illustrate the importance of the data-driven closure terms in the NAR model, which account for the effects of the unresolved dynamics brought by nonlinear interactions.