Viscous Burgers Equation Research Articles

Solutions to viscous Burgers equations with time dependent source term

We study the existence and uniqueness of weak solutions for a Cauchy problem of a viscous Burgers equation with a time dependent reaction term involving Dirac measure. After applying a Hopf like transformation, we investigate the associated two initial boundary value problems by assuming a common boundary. The existence of the boundary data is shown with the help of Abel's integral equation. We then derive explicit representation of the boundary function. Also, we prove that the solutions of associated initial boundary value problems converge uniformly to a nonzero constant on compact sets as t approaches infinity.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/02/abstr.html

Parameterized nonintrusive reduced‐order model for general unsteady flow problems using artificial neural networks

AbstractA nonintrusive reduced‐order model for nonlinear parametric flow problems is developed. It is based on extracting a reduced‐order basis from high‐order snapshots via proper orthogonal decomposition and using multilayered feedforward artificial neural networks to approximate the reduced‐order coefficients. The model is a generic and efficient approach for the reduction of time‐dependent parametric systems, including those described by partial differential equations. Since it is nonintrusive, it is independent of the high‐order computational method and can be used together with black‐box solvers. Numerical studies are presented for steady‐state isentropic nozzle flow with geometric parameterization and unsteady parameterized viscous Burgers equation. An adaptive sampling strategy is proposed to increase the quality of the neural network approximation while minimizing the required number of parameter samples and, as a direct consequence, the number of high‐order snapshots and the size of the network training set. Results confirm the accuracy of the nonintrusive approach as well as the speed‐up achieved compared with intrusive hyper‐reduced‐order approaches.

Numerical simulation of the coupled viscous Burgers equation using the Hermite formula and cubic B-spline basis functions

A numerical procedure dependent on the cubic B-spline and the Hermite formula is developed for the coupled viscous Burgers’ equation (CVBE). The method uses a combination of the Hermite formula and the cubic B-spline for discretization of the space dimension while the time dimension is approximated using the typical finite differences. A piecewise continuous sufficiently smooth function is obtained as a solution which allows to approximate solution at any location in the domain of interest. The scheme is tested for stability analysis and is proved to be unconditionally stable. Numerical experiments and comparison of outcomes reveal that the suggested scheme comes up with better accuracy and is extremely productive.

Forward sensitivity approach for estimating eddy viscosity closures in nonlinear model reduction.

In this paper, we propose a variational approach to estimate eddy viscosity using forward sensitivity method (FSM) for closure modeling in nonlinear reduced order models. FSM is a data assimilation technique that blends model's predictions with noisy observations to correct initial state and/or model parameters. We apply this approach on a projection based reduced order model (ROM) of the one-dimensional viscous Burgers equation with a square wave defining a moving shock, and the two-dimensional vorticity transport equation formulating a decay of Kraichnan turbulence. We investigate the capability of the approach to approximate an optimal value for eddy viscosity with different measurement configurations. Specifically, we show that our approach can sufficiently assimilate information either through full-field or sparse noisy measurements to estimate eddy viscosity closure to cure standard Galerkin reduced order model (GROM) predictions. Therefore, our approach provides a modular framework to correct forecasting error from a sparse observational network on a latent space. We highlight that the proposed GROM-FSM framework is promising for emerging digital twin applications, where real-time sensor measurements can be used to update and optimize surrogate model's parameters.

Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism

Model reduction methods aim to describe complex dynamic phenomena using only relevant dynamical variables, decreasing computational cost, and potentially highlighting key dynamical mechanisms. In the absence of special dynamical features such as scale separation or symmetries, the time evolution of these variables typically exhibits memory effects. Recent work has found a variety of data-driven model reduction methods to be effective for representing such non-Markovian dynamics, but their scope and dynamical underpinning remain incompletely understood. Here, we study data-driven model reduction from a dynamical systems perspective. For both chaotic and randomly-forced systems, we show the problem can be naturally formulated within the framework of Koopman operators and the Mori-Zwanzig projection operator formalism. We give a heuristic derivation of a NARMAX (Nonlinear Auto-Regressive Moving Average with eXogenous input) model from an underlying dynamical model. The derivation is based on a simple construction we call Wiener projection, which links Mori-Zwanzig theory to both NARMAX and to classical Wiener filtering. We apply these ideas to the Kuramoto-Sivashinsky model of spatiotemporal chaos and a viscous Burgers equation with stochastic forcing.

New cubic B-spline approximation technique for numerical solutions of coupled viscous Burgers equations

PurposeThe purpose of this paper is to present a new cubic B-spline (CBS) approximation technique for the numerical treatment of coupled viscous Burgers’ equations arising in the study of fluid dynamics, continuous stochastic processes, acoustic transmissions and aerofoil flow theory.Design/methodology/approachThe system of partial differential equations is discretized in time direction using the finite difference formulation, and the new CBS approximations have been used to interpolate the solution curves in the spatial direction. The theoretical estimation of stability and uniform convergence of the proposed numerical algorithm has been derived rigorously.FindingsA different scheme based on the new approximation in CBS functions is proposed which is quite different from the existing methods developed (Mittal and Jiwari, 2012; Mittal and Arora, 2011; Mittal and Tripathi, 2014; Raslan et al., 2017; Shallal et al., 2019). Some numerical examples are presented to validate the performance and accuracy of the proposed technique. The simulation results have guaranteed the superior performance of the presented algorithm over the existing numerical techniques on approximate solutions of coupled viscous Burgers’ equations.Originality/valueThe current approach based on new CBS approximations is novel for the numerical study of coupled Burgers’ equations, and as far as we are aware, it has never been used for this purpose before.

On the viscous Burgers equation on metric graphs and fractals

We study a formulation of Burgers equation on the Sierpinski gasket, which is the prototype of a p.c.f. self-similar fractal. One possibility is to implement Burgers equation as a semilinear heat equation associated with the Laplacian for scalar functions, just as on the unit interval. Here we propose a second, different formulation which follows from the Cole-Hopf transform and is associated with the Laplacian for vector fields. The difference between these two equations can be understood in terms of different vertex conditions for Laplacians on metric graphs. For the second formulation we show existence and uniqueness of solutions and verify the continuous dependence on the initial condition. We also prove that solutions on the Sierpinski gasket can be approximated in a weak sense by solutions to corresponding equations on approximating metric graphs. These results are part of a larger program discussing nonlinear partial differential equations on fractal spaces.

Remark on the global null controllability for a viscous Burgers-particle system with particle supported control

This paper is devoted to study the controllability of a one-dimensional fluid-particle interaction model where the fluid follows the viscous Burgers equation and the point mass obeys Newton’s second law. We prove the null controllability for the velocity of the fluid and the particle and an approximate controllability for the position of the particle with a control variable acting only on the particle. One of the novelties of our work is the fact that we achieve this controllability result in a uniform time for all initial data and without any smallness assumptions on the initial data.

Time-series learning of latent-space dynamics for reduced-order model closure

We study the performance of long short-term memory networks (LSTMs) and neural ordinary differential equations (NODEs) in learning latent-space representations of dynamical equations for an advection-dominated problem given by the viscous Burgers equation. Our formulation is devised in a nonintrusive manner with an equation-free evolution of dynamics in a reduced space with the latter being obtained through a proper orthogonal decomposition. In addition, we leverage the sequential nature of learning for both LSTMs and NODEs to demonstrate their capability for closure in systems that are not completely resolved in the reduced space. We assess our hypothesis for two advection-dominated problems given by the viscous Burgers equation. We observe that both LSTMs and NODEs are able to reproduce the effects of the absent scales for our test cases more effectively than does intrusive dynamics evolution through a Galerkin projection. This result empirically suggests that time-series learning techniques implicitly leverage a memory kernel for coarse-grained system closure as is suggested through the Mori–Zwanzig formalism.

Resource-Constrained Model Selection for Uncertainty Propagation and Data Assimilation

All observable phenomena can be described by alternative mathematical models, which vary in their fidelity and computational cost. Selection of an appropriate model involves a tradeoff between computational cost and representational accuracy. Ubiquitous uncertainty (randomness) in model parameters and forcings, and assimilation of observations of the system states into predictions, complicate the model selection problem. We present a framework for analysis of the impact of data assimilation on cost-constrained model selection. The framework relies on the definitions of cost and accuracy functions in the context of data assimilation for multifidelity models with uncertain (random) coefficients. It contains an estimate of error bounds for a system's state prediction obtained by assimilating data into a model via an ensemble Kalman filter. This estimate is given in terms of model error, sampling error, and data error. Two examples illustrating the applicability of our model selection method are provided. The first example deals with an ordinary differential equation, for which a sequence of lower-fidelity models is constructed by progressively increasing the time step used in its discretization. The second example comprises the viscous Burgers equation as the high-fidelity model and a linear advection-diffusion equation as its low-fidelity counterpart.

An adaptive WENO algorithm for one-dimensional convection-dominated partial differential equations

In this work, a versatile numerical algorithm was proposed for solving one-dimensional nonlinear convection-dominated partial differential equations as these found in the simulation of diverse chemical engineering applications. The proposed algorithm uses a fully adaptive scheme, in which both the grid spacing and the time discretization are dynamically adjusted. It uses a finite volume discretization with a variable number of grid cells and an explicit time integration with time-step control. The high-order Weighted Essentially Non-Oscillatory (WENO) scheme on non-uniform grid was combined with a grid refinement technique based on the equitable distribution principle and a spatial regularization procedure. The time-stepping procedure as well as some implementation issues are minutely discussed. The capability and efficiency of the new algorithm was demonstrated through the numerical simulation of five chemical engineering applications: the Viscous Burgers Equation, a Chromatographic column, the Buckley-Leverett Equation, and two examples of reaction-convection systems. For all analyzed cases, it was verified that the number of grid cells required to capture steep gradients can be greatly reduced with the proposed grid adaptation scheme.

A comparison between Cole-Hopf Tranformation and Homotopy Perturbation Method for Viscous Burger Equation in Traffic Flow

This paper presents the comparison of Cole-Hopf transformation and Homotopy Perturbation Method for solving viscous Burgers Equation in traffic flow problems. A problem is presented to demonstrate the differences. The result shows that even though Cole-Hopf transformation brings exact solution for all cases, the formula is in complicated form. Compared to this, Homotopy Perturbation Method is more effective to solve burger equation with boundary condition in traffic flow problems.

Formation of viscous fingers in regularized Laplacian growth.

A systematic analytic treatment of local fluctuations in the regularized Laplacian growth problem is given. The interface dynamics is stabilized by a short-distance cutoff ℏ preventing the cusps production in a finite time. The regularization mechanism results in the violation of the incompressibility condition of the viscous fluid on a microscale in the vicinity of the moving interface, thus producing local fluctuations of pressure. Dissipation of fluctuations with time is described by universal Dyson Brownian motion, which reduces to the complex viscous Burgers equation in the hydrodynamic approximation. Because of the intrinsic instability of the interface dynamics, tiny fluctuations of pressure generate universal complex patterns with well developed fjords and fingers in a long time asymptotic.

Generalized Burgers equation with rough transport noise

We introduce a new technique for studying well posedness and energy estimates for evolution equations with a rough transport term. The technique is based on finding suitable space–time weight functions for the equations at hand. As an example we study the well posedness of the generalized viscous Burgers equation perturbed by a rough path transport noise.

An artificial neural network framework for reduced order modeling of transient flows

This paper proposes a supervised machine learning framework for the non-intrusive model order reduction of unsteady fluid flows to provide accurate predictions of non-stationary state variables when the control parameter values vary. Our approach utilizes a training process from full-order scale direct numerical simulation data projected on proper orthogonal decomposition (POD) modes to achieve an artificial neural network (ANN) model with reduced memory requirements. This data-driven ANN framework allows for a nonlinear time evolution of the modal coefficients without performing a Galerkin projection. Our POD-ANN framework can thus be considered an equation-free approach for latent space dynamics evolution of nonlinear transient systems and can be applied to a wide range of physical and engineering applications. Within this framework we introduce two architectures, namely sequential network (SN) and residual network (RN), to train the trajectory of modal coefficients. We perform a systematic analysis of the performance of the proposed reduced order modeling approaches on prediction of a nonlinear wave-propagation problem governed by the viscous Burgers equation, a simplified prototype setting for transient flows. We find that the POD-ANN-RN yields stable and accurate results for test problems assessed both within inside and outside of the database range and performs significantly better than the standard intrusive Galerkin projection model. Our results show that the proposed framework provides a non-intrusive alternative to the evolution of transient physics in a POD basis spanned space, and can be used as a robust predictive model order reduction tool for nonlinear dynamical systems.

Uncertainty quantification methodology for hyperbolic systems with application to blood flow in arteries

We present a Stochastic Finite Volume - ADER (SFV-ADER) methodology for Uncertainty Quantification (UQ) in the general framework of systems of hyperbolic balance laws with uncertainty in parameters. The resulting method is weakly intrusive, meaning that deterministic solvers need minor modifications to include random contributions, and has no theoretical accuracy barrier. An illustration of the second-order version for the viscous Burgers equation with uncertain viscosity coefficient is first given in detail, under different deterministic initial conditions; the attainment of the theoretically expected convergence rate is demonstrated empirically; results and features of our scheme are discussed. We then extend the SFV-ADER method to a non-linear hyperbolic system with source terms that models one-dimensional blood flow in arteries, assuming uncertainties in a set of parameters of the problem. Results are then compared with measurements in a well-defined 1:1 experimental replica of the network of largest arteries in the human systemic circulation. For a 99%-confidence level, severe influence of parameter fluctuations is seen in pressure profiles, while secondary effects on flow rate profiles become visible, particularly in terminal branches. Results suggest that the proposed methodology could successfully be applied to other physical problems involving hyperbolic balance laws.

Analytical solutions to a nonlinear diffusion–advection equation

In this paper, we construct some exact and analytical solutions to a nonlinear diffusion and advection model (Pudasaini in Eng Geol 202: 62–73, 2016) using the Lie symmetry, travelling wave, generalized separation of variables, and boundary layer methods. The model in consideration can be viewed as an extension of viscous Burgers equation, but it describes significantly different physical phenomenon. The nonlinearity in the model is associated with the quadratic diffusion and advection fluxes which are described by the sub-diffusive and sub-advective fluid flow in general porous media and debris material. We also observe that different methods consistently produce similar analytical solutions. This highlights the intrinsic characteristics of the flow of fluid in porous material. The nonlinear diffusion and advection is characterized by a gradually thinning tail that stretches to the rear of the fluid and the evolution of forward advecting frontal bore head, in contrast to the classical linear diffusion and advection. Additionally, we compare solutions for the linear and nonlinear diffusion and advection models highlighting the similarities and differences. The analytical solutions constructed in this paper and the existing high-resolution numerical solution presented previously for the nonlinear diffusion and advection model independently support each other. This implies that the exact and analytical solutions constructed here are physically meaningful and can potentially be applied to calculate the complex nonlinear re-distribution of fluid in porous landscape, and debris and porous materials.

Convection–diffusion equations with random initial conditions

We consider an evolution equation generalising the viscous Burgers equation supplemented by an initial condition which is a homogeneous random field. We develop a non-linear framework enabling us to show the existence and regularity of solutions as well as their long time behaviour.

On boundary layers for the Burgers equations in a bounded domain

As a simplified model derived from the Navier–Stokes equations, we consider the viscous Burgers equations in a bounded domain with two-point boundary conditions. We investigate the singular behaviors of their solutions uε as the viscosity parameter ε gets smaller. The idea is constructing the asymptotic expansions in the order of the ε and validating the convergence of the expansions to the solutions uε as ε → 0. In this article, we consider the case where sharp transitions occur at the boundaries, i.e. boundary layers, and we fully analyze the convergence at any order of ε using the so-called boundary layer correctors. We also numerically verify the convergences.

Algebraic traveling waves for the generalized viscous Burgers equation

In this paper we apply the general results in [10] on algebraic traveling wave solutions for n-th order nonlinear evolution equations in one space dimension to prove a full classification of the algebraic traveling wave solutions of a second order generalized viscous Burgers' equation.