One-dimensional Viscous Burgers Research Articles

Transverse modulational dynamics of quenched patterns.

We study the modulational dynamics of striped patterns formed in the wake of a planar directional quench. Such quenches, which move across a medium and nucleate pattern-forming instabilities in their wake, have been shown in numerous applications to control and select the wavenumber and orientation of striped phases. In the context of the prototypical complex Ginzburg-Landau and Swift-Hohenberg equations, we use a multiple-scale analysis to derive a one-dimensional viscous Burgers' equation, which describes the long-wavelength modulational and defect dynamics in the direction transverse to the quenching motion, that is, along the quenching line. We show that the wavenumber selecting properties of the quench determine the nonlinear flux parameter in the Burgers' modulation equation, while the viscosity parameter of the Burgers' equation is naturally determined by the transverse diffusivity of the pure stripe state. We use this approximation to accurately characterize the transverse dynamics of several types of defects formed in the wake, including grain boundaries and phase-slips.

A multifidelity deep operator network approach to closure for multiscale systems

Projection-based reduced order models (PROMs) have shown promise in representing the behavior of multiscale systems using a small set of generalized (or latent) variables. Despite their success, PROMs can be susceptible to inaccuracies, even instabilities, due to the improper accounting of the interaction between the resolved and unresolved scales of the multiscale system (known as the closure problem). In the current work, we interpret closure as a multifidelity problem and use a multifidelity deep operator network (DeepONet) framework to address it. In addition, to enhance the stability and accuracy of the multifidelity-based closure, we employ the recently developed “in-the-loop” training approach from the literature on coupling physics and machine learning models. The resulting approach is tested on shock advection for the one-dimensional viscous Burgers equation and vortex merging using the two-dimensional Navier–Stokes equations. The numerical experiments show significant improvement of the predictive ability of the closure-corrected PROM over the un-corrected one both in the interpolative and the extrapolative regimes.

H-adaptation for high-order discontinuous Galerkin schemes built on local multiwavelet analysis

We develop and analyze error estimators and mesh adaptation strategies within a discontinuous Galerkin formulation. The basic idea of the study is to reduce the computational cost of the simulation by employing mesh adaptation as a better alternative to the use of uniform grids. The novelty of the study resides in the use of multiwavelets and how their remarkable properties may shed new light on driving the adaptation process. This is motivated by the fact that multiwavelets break any input apart into a hierarchy of low resolution data and subsequently finer details. Our error estimator makes use of multiwavelets’ properties while being local to the element, thereby maintaining the parallel efficiency of the solver. Early tests on the one-dimensional viscous Burgers equation have shown convincing results (García Bautista et al. [1]). This work is focused on the laminar backward-facing step configuration to assess the performance of the method in higher dimensions.

Stochastic turbulence for Burgers equation driven by cylindrical Lévy process

This work is devoted to investigating stochastic turbulence for the fluid flow in one-dimensional viscous Burgers equation perturbed by Lévy space-time white noise with the periodic boundary condition. We rigorously discuss the regularity of solutions and their statistical quantities in this stochastic dynamical system. The quantities include moment estimate, structure function and energy spectrum of the turbulent velocity field. Furthermore, we provide qualitative and quantitative properties of the stochastic Burgers equation when the kinematic viscosity [Formula: see text] tends towards zero. The inviscid limit describes the strong stochastic turbulence.

Numerical solution and bifurcation analysis of nonlinear partial differential equations with extreme learning machines

We address a new numerical method based on a class of machine learning methods, the so-called Extreme Learning Machines (ELM) with both sigmoidal and radial-basis functions, for the computation of steady-state solutions and the construction of (one-dimensional) bifurcation diagrams of nonlinear partial differential equations (PDEs). For our illustrations, we considered two benchmark problems, namely (a) the one-dimensional viscous Burgers with both homogeneous (Dirichlet) and non-homogeneous boundary conditions, and, (b) the one- and two-dimensional Liouville–Bratu–Gelfand PDEs with homogeneous Dirichlet boundary conditions. For the one-dimensional Burgers and Bratu PDEs, exact analytical solutions are available and used for comparison purposes against the numerical derived solutions. Furthermore, the numerical efficiency (in terms of numerical accuracy, size of the grid and execution times) of the proposed numerical machine-learning method is compared against central finite differences (FD) and Galerkin weighted-residuals finite-element (FEM) methods. We show that the proposed numerical machine learning method outperforms in terms of numerical accuracy both FD and FEM methods for medium to large sized grids, while provides equivalent results with the FEM for low to medium sized grids; both methods (ELM and FEM) outperform the FD scheme. Furthermore, the computational times required with the proposed machine learning scheme were comparable and in particular slightly smaller than the ones required with FEM.

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.

Numerical efficiency of some exponential methods for an advection–diffusion equation

ABSTRACTIn this paper, we investigate several modified exponential finite-difference methods to approximate the solution of the one-dimensional viscous Burgers' equation. Burgers' equation admits solutions that are positive and bounded under appropriate conditions. Motivated by these facts, we propose nonsingular exponential methods that are capable of preserving some structural properties of the solutions of Burgers' equation. The fact that some of the techniques preserve structural properties of the solutions is thoroughly established in this work. Rigorous analyses of consistency, stability and numerical convergence of these schemes are presented for the first time in the literature, together with estimates of the numerical solutions. The methods are computationally improved for efficiency using the Padé approximation technique. As a result, the computational cost is substantially reduced in this way. Comparisons of the numerical approximations against the exact solutions of some initial-boundary-value problems for different Reynolds numbers show a good agreement between them.

Resolving controls for exact and approximate controllability of viscous Burgers’ equation: The Green’s function approach

In this paper, we consider a nonlinear control problem for one-dimensional viscous Burgers’ equation associated with a controlled linear heat equation by means of the Hopf–Cole transformation. The control is carried out by the time-dependent intensity of a distributed heat source influencing the heat equation. The set of admissible controls consists of compactly supported [Formula: see text] functions. Using the Green’s function approach, we analyze the possibilities of exact and approximate establishment of a given terminal state for the associated nonlinear Burgers’ equation within a desired amount of time. It is shown that the exact controllability of the associated Burgers’ equation and the heat equation are equivalent. Furthermore, sufficient conditions for the approximate controllability are derived. The set of resolving controls is constructed in both cases. The determination of the resolving controls providing exact controllability is reduced to an infinite-dimensional system of linear algebraic equations. By means of the heuristic method of resolving control determination, parametric hierarchies of solutions providing approximate controllability are constructed. The results of a numerical simulation supporting the theoretical derivations are discussed.

Large time behavior of solutions to the compressible Navier–Stokes equations around periodic steady states

This paper shows that the strong solution to the compressible Navier–Stokes equation around spatially periodic stationary solution in a periodic layer of Rn(n=2,3) exists globally in time if Reynolds and Mach numbers are sufficiently small. It is proved that the asymptotic leading part of the perturbation is given by a solution to the one-dimensional viscous Burgers equation multiplied by a spatially periodic function when n=2, and by a solution to the two-dimensional heat equation multiplied by a spatially periodic function when n=3.

Existence of Globally Attracting Solutions for One-Dimensional Viscous Burgers Equation with Nonautonomous Forcing---A Computer Assisted Proof

We prove the existence of globally attracting solutions of the viscous Burgers equation with periodic boundary conditions on the line for some particular choices of viscosity and non-autonomous forcing. The attract- ing solution is periodic if the forcing is periodic. The method is general and can be applied to other similar partial differential equations. The proof is computer assisted.

Optimal spatiotemporal reduced order modeling of the viscous Burgers equation

The one-dimensional viscous Burgers equation with a time-periodic inflow boundary condition is investigated within the context of a newly developed optimal spatiotemporal reduced order modeling (OPSTROM) framework. Flow simulations are carried out with a conventional finite-difference scheme, and are expedited by coarsening the computational grid in space and time. The OPSTROM framework is used to maintain reliable predictions for the flow by constructing interactive subgrid-scale models to account for the effects due to unresolved spatial and temporal scales. Model construction is data-driven, and is based upon principles of mean-square error minimization, conditional expectations and stochastic estimation. The results indicate a need to model both subgrid spatial and temporal scales in order to improve the accuracy of under-resolved simulations.

Modified Burgers' equation by the local discontinuous Galerkin method

In this paper, we present the local discontinuous Galerkin method for solving Burgers' equation and the modified Burgers' equation. We describe the algorithm formulation and practical implementation of the local discontinuous Galerkin method in detail. The method is applied to the solution of the one-dimensional viscous Burgers' equation and two forms of the modified Burgers' equation. The numerical results indicate that the method is very accurate and efficient.

Asymptotic Behavior of Solutions to the Compressible Navier--Stokes Equation Around a Time-Periodic Parallel Flow

The global in time existence of strong solutions to the compressible Navier--Stokes equation around time-periodic parallel flows in $\mathbb{R}^n$, $n\geq 2$, is established under smallness conditions on Reynolds number, Mach number, and initial perturbations. Furthermore, it is proved for $n=2$ that the asymptotic leading part of solutions is given by a solution of the one-dimensional viscous Burgers equation multiplied by the time-periodic function. In the case $n\geq 3$ the asymptotic leading part of solutions is given by a solution of the $n-1$-dimensional heat equation with the convective term multiplied by the time-periodic function.

Study of an entropy-consistent Navier–Stokes flux

This paper presents a study to achieve discrete entropy consistency using artificial and physical diffusion mechanisms. The study begins with the one-dimensional viscous Burgers equation, specifically looking at the shock results of entropy-conserved fluxes combined with a few choices of artificial and physical viscous diffusions. The approach is then repeated for the Navier–Stokes equations. Overall, it is demonstrated that the artificial viscosity (or entropy) terms are still needed in addition to physical viscosity to achieve entropy consistency in shock predictions, although one of the artificial terms can be dropped for high viscosity or low Reynolds number flow.

Asymptotic Behavior of Solutions to the Compressible Navier–Stokes Equation Around a Parallel Flow

The initial boundary value problem for the compressible Navier–Stokes equation is considered in an infinite layer of \({\mathbf{R}^{2}}\) . It is proved that if the Reynolds and Mach numbers are sufficiently small, then strong solutions to the compressible Navier–Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations. The large time behavior of the solution is described by a solution of a one-dimensional viscous Burgers equation. The proof is given by a combination of spectral analysis of the linearized operator and a variant of the Matsumura–Nishida energy method.

Gain-scheduled controller synthesis for a nonlinear PDE

Linear parameter-varying (LPV) modelling and control of a nonlinear partial differential equation (PDE) is considered in this article. The one-dimensional viscous Burgers' equation is discretised using a finite difference scheme; the boundary conditions are taken as control inputs and the velocities at two grid points are assumed to be measurable. A nonlinear high-order state space model is generated and proper orthogonal decomposition is used for model order reduction. After assessing the accuracy of the reduced model, a low-order functional observer is designed to estimate the reduced states which are linear combinations of the velocities at all grid points. A discrete-time quasi-LPV model that is affine in scheduling parameters is derived based on the reduced model. A polytopic LPV controller is synthesised based on a generalised plant containing the LPV model and the functional observer. More generally, the proposed method can be used to design an LPV controller for a quasi-LPV system with non-measurable scheduling parameters. Simulation results demonstrate the high tracking performance and disturbance and measurement noise rejection capabilities of the designed LPV controller compared with a linear quadratic Gaussian (LQG) controller based on a linearised model.

LPV Modelling and Control of Burgers’ Equation

AbstractLinear parameter-varying (LPV) modelling and control of a nonlinear PDE is presented in this paper. The one-dimensional viscous Burgers’ equation is discretized using a finite difference scheme and the boundary conditions are taken as control inputs. A nonlinear high-order state space model is generated and proper orthogonal decomposition is used for model order reduction and the accuracy of the reduced model is verified. A discrete-time quasi-LPV model that is affine in scheduling parameters is derived based on the reduced model and a polytopic LPV controller is synthesized. A low-order functional observer is designed to estimate the scheduling parameters required for LPV controller. Simulation results demonstrate the high tracking performance and disturbance and measurement noise rejection capabilities of the designed LPV controller comparing with an LQG controller based on a linearized model.

A New Diffusion Scheme for Numerical Models Based on Full Irreversibility

Abstract In this work, the forecast accuracy of a numerical weather prediction model is improved by emulating physical dissipation as suggested by the second law of thermodynamics, which controls the irreversible evolutionary direction of a many-body system like the atmosphere. The ability of the new physics-based scheme to improve model accuracy is demonstrated via the case of the one-dimensional viscous Burgers equation and the one-dimensional diffusion equation, as well as a series of numerical simulations of the well-known 1998 successive torrential rains along the Yangtze River valley and 365 continuous 24-h simulations during 2005–06 with decreased root-mean-square errors and improved forecasts in all of the simulations.

De-aliasing on non-uniform grids: algorithms and applications

We present de-aliasing rules to be used when evaluating non-linear terms with polynomial spectral methods on non-uniform grids analogous to the de-aliasing rules used in Fourier spectral methods. They are based upon the idea of super-collocation followed by a Galerkin projection of the non-linear terms. We demonstrate through numerical simulation that both accuracy and stability can be greatly enhanced through the use of this approach. We begin by deriving from the numerical quadrature rules used by Galerkin-type projection methods the number of quadrature points and weights needed for quadratic and cubic non-linearities. We then present a systematic study of the effects of super-collocation when using both a continuous Galerkin and a discontinuous Galerkin method to solve the one-dimensional viscous Burgers equation. We conclude by examining three direct numerical simulation flow examples: incompressible turbulent flow in a triangular duct, incompressible turbulent flow in a channel at Re τ =395, and compressible flow past a pitching airfoil at Re=45,000.

An accurate integral‐based scheme for advection–diffusion equation

AbstractThis paper proposes an accurate integral‐based scheme for solving the advection–diffusion equation. In the proposed scheme the advection–diffusion equation is integrated over a computational element using the quadratic polynomial interpolation function. Then elements are connected by the continuity of first derivative at boundary points of adjacent elements. The proposed scheme is unconditionally stable and results in a tridiagonal system of equations which can be solved efficiently by the Thomas algorithm. Using the method of fractional steps, the proposed scheme can be extended straightforwardly from one‐dimensional to multi‐dimensional problems without much difficulty and complication. To investigate the computational performances of the proposed scheme five numerical examples are considered: (i) dispersion of Gaussian concentration distribution in one‐dimensional uniform flow; (ii) one‐dimensional viscous Burgers equation; (iii) pure advection of Gaussian concentration distribution in two‐dimensional uniform flow; (iv) pure advection of Gaussian concentration distribution in two‐dimensional rigid‐body rotating flow; and (v) three‐dimensional diffusion in a shear flow. In comparison not only with the QUICKEST scheme, the fully time‐centred implicit QUICK scheme and the fully time‐centred implicit TCSD scheme for one‐dimensional problem but also with the ADI‐QUICK scheme, the ADI‐TCSD scheme and the MOSQUITO scheme for two‐dimensional problems, the proposed scheme shows convincing computational performances. Copyright © 2001 John Wiley & Sons, Ltd.