Nonlinear Viscous Burgers Research Articles

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.

Two-stage explicit schemes based on numerical approximations of convection-diffusion equations

A wide family of finite difference methods for the convection-diffusion problems based on an explicit two-stage scheme and a seemingly implicit method are presented. In this paper, to have a greater stability region while maintaining the second-order accuracy, a family of methods that combines the MacCormack and Saul'vey schemes is proposed. The stability analysis of the combined methods is investigated using the von Neumann approach. In each case, it is found that it is the convection term that limits the stability of the scheme. Based on the von Neumann analysis, valuable stability limits in terms of mesh parameters for maintaining accurate results are determined in an analytic manner and demonstrated through computer simulations. Two model problems consisting of linear advection-diffusion and nonlinear viscous Burgers equation are given to illustrate some properties of the present technique, such as stability and the ability to propagate discontinuities.

Efficient computation of 1D and 2D nonlinear Viscous Burgers’ equation

Model order reduction (MOR) has emerged as a de-facto solution for simulation, control, and optimization of large-scale and complex dynamical systems from the past decades. This paper demonstrates two comprehensive MOR approaches to overcome the overall computational cost required to simulate the nonlinear, viscous Burger’s equation. The former scheme is the approximate proper orthogonal decomposition (APOD), which involves capturing the approximate snapshot ensemble of the states by performing multiple linearizations along the state-trajectory. The latter technique is based on the notion of nonlinear moment matching (NLMM), in which the reduced projection basis is generated by numerically solving the simplified, nonlinear, Sylvester equation. Both the reduction schemes are tested on the high-fidelity finite difference (FD) formulation of 1D and the 2D Burgers’ equation. The results show substantial computational savings in extracting the orthonormal projection basis required to generate the reduced manifold. Furthermore, a comparative study of both methods is presented with proper orthogonal decomposition (POD) for various test signals.

Improving the precision of discrete numerical solutions using the generalized integral transform technique

Most spatial discretizations applied by finite difference and volume methods to advective/convective terms introduce significant diffusive and dispersive numerical errors, where the former is often required to improve the numerical stability to the resulting scheme. This paper presents a new approach that employs integral transforms to improve the accuracy of available discrete numerical solutions. The method uses a known and inaccurate numerical solution to filter the original governing equations, solves the resulting problem using integral transforms, and then uses it to correct the original numerical solution. Before doing so, these numerical solutions are rewritten as a series representation, which is done using the same eigenfunction basis employed by the integral transformation. Different test cases, based on the unsteady and one-dimensional wave motion described by the nonlinear viscous Burgers’ equation in a finite domain, are selected to evaluate the proposed approach. These test cases involve numerical solutions with different types of errors, and are compared with a highly accurate numerical solution for verification. The integral transform technique, in combination with different numerical filters, showed that inaccurate numerical solutions can in fact be corrected by this methodology, as very satisfactory errors were generally obtained. This is true for smooth enough numerical solutions, i.e., the procedure is not advisable to problems that lead to discontinuous numerical solutions.

Efficient Chebyshev Pseudospectral Methods for Viscous Burgers’ Equations in One and Two Space Dimensions

Spectral methods are powerful numerical methods used for the solution of ordinary and partial differential equations. In this paper, we propose an efficient and accurate numerical method for the one and two dimensional nonlinear viscous Burgers’ equations and coupled viscous Burgers’ equations with various values of viscosity subject to suitable initial and boundary conditions. The method is based on the Chebyshev collocation technique in space and the fourth-order Runge–Kutta method in time. This proposed scheme is robust, fast, flexible, and easy to implement using modern mathematical software such as Matlab. Furthermore, it can be easily modified to handle other problems. Extensive numerical simulations are presented to demonstrate the accuracy of the proposed scheme. The numerical solutions for different values of the Reynolds number are compared with analytical solutions as well as other numerical methods available in the literature. Compared to other numerical methods, the proposed method is shown to have higher accuracy with fewer nodes. Our numerical experiments show that the Chebyshev collocation method is an efficient and reliable scheme for solving Burgers’ equations with reasonably high Reynolds number.

A Single Sweep AGE Algorithm based on Off-Step Discretization for the Solution of Viscous Burgers’ Equation on a Variable Mesh

In this paper, we discuss a new single sweep alternating group explicit iteration method, along with a third order numerical method based on off-step discretization on a variable mesh to solve nonlinear viscous Burgers’ equation subject to given natural boundary conditions. The proposed method is also applicable to both singular and non-singular problems, which is main attraction of our work. The convergence of the proposed method is discussed in detail. We compared the results of proposed iteration method with the results of corresponding double sweep alternating group explicit iteration method to demonstrate computationally the efficiency of the proposed method.

SWAGE algorithm for the cubic spline solution of nonlinear viscous Burgers’ equation on a geometric mesh

In this paper, we discuss the single sweep alternating group explicit (SWAGE) and Newton-SWAGE iteration methods to solve the non-linear ordinary differential equation y″=f(x,y,y′) subject to given natural boundary conditions, along with a third order cubic spline numerical method on a geometric mesh. It is applicable to both singular and non-singular problems. The convergence of the SWAGE iteration method is discussed in detail. We compared the results of proposed SWAGE iteration method with the results of corresponding two parameter alternating group explicit (TAGE) iteration methods to demonstrate computationally the efficiency of the proposed method.

A Volterra series approach to the frequency domain analysis of non-linear viscous Burgers’ equation

In this paper, higher order frequency response functions, based on the Volterra series, are employed to characterise the input-output behaviour of the non-linear viscous Burgers’ equation subject to sinusoidal excitation. First, a formal Volterra series representation for each spatial location is derived for the solution of Burgers’ equation with a boundary condition as the input to the system. Then a systematic method is presented to obtain the higher order frequency kernels of the Volterra series at each spatial location by solving a series of ordinary differential equations. It is shown that the convergence region of the individual harmonics with respect to the magnitude of the input excitation can be estimated by using these higher order kernels. The frequency characteristics of Burgers’ equation is investigated and compared with numerical simulation.

A comparative study of the performance of high resolution advection schemes in the context of data assimilation

High resolution advection schemes have been developed and studied to model propagation of flows involving sharp fronts and shocks. So far the impact of these schemes in the framework of inverse problem solution has been studied only in the context of linear models. A detailed study of the impact of various slope limiters and the piecewise parabolic method (PPM) on data assimilation is the subject of this work, using the nonlinear viscous Burgers equation in 1−D. Also provided are results obtained in 2−D using a global shallow water equations model. The results obtained in this work may point out to suitability of these advection schemes for data assimilation in more complex higher dimensional models. Copyright © 2005 John Wiley & Sons, Ltd.