1D Burgers Research Articles

Second-order large time step wave adding scheme for hyperbolic conservation laws

This paper constructs a second-order large time step wave adding scheme (LTS-WA2) for hyperbolic conservation laws. Based on the first-order large time step wave adding scheme (LTS-WA1), a piece-wise linear reconstruction with limiter is performed on the solutions, and the band decomposition and band adding is complemented (into the discontinuity decomposition and wave adding), then the scheme is extended to second-order. The manufacture of the new scheme is simplified and written uniformly. Theoretical analyses are made for scalar cases. Numerical experiments give 11 tests which involve 1D scalar equations, 1D, 2D and 3D Euler equations. Computations show that the new scheme maintains high resolution and low dissipation of the original scheme at large CFL number, and also improves the problem of low resolution of the original scheme for rarefaction wave and contact discontinuity at small CFL number (roughly less than 2). Accuracy order is analyzed theoretically and validated using tests of 1D linear equation and 2D Burgers equation. CPU time is also compared with traditional second-order (Harten-TVD) scheme, showing that LTS-WA2 is more cost effective.

Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks

In recent years, deep learning has proven to be a viable methodology for surrogate modeling and uncertainty quantification for a vast number of physical systems. However, in their traditional form, such models can require a large amount of training data. This is of particular importance for various engineering and scientific applications where data may be extremely expensive to obtain. To overcome this shortcoming, physics-constrained deep learning provides a promising methodology as it only utilizes the governing equations. In this work, we propose a novel auto-regressive dense encoder-decoder convolutional neural network to solve and model non-linear dynamical systems without training data at a computational cost that is potentially magnitudes lower than standard numerical solvers. This model includes a Bayesian framework that allows for uncertainty quantification of the predicted quantities of interest at each time-step. We rigorously test this model on several non-linear transient partial differential equation systems including the turbulence of the Kuramoto-Sivashinsky equation, multi-shock formation and interaction with 1D Burgers' equation and 2D wave dynamics with coupled Burgers' equations. For each system, the predictive results and uncertainty are presented and discussed together with comparisons to the results obtained from traditional numerical analysis methods.

Improved rainfall nowcasting using Burgers’ equation

Nowcasting of surface precipitation from radar data typically relies on algorithms that calculate advection, such as the McGill Algorithm for Precipitation nowcasting by Lagrangian Extrapolation (MAPLE). This method offers high spatial and temporal resolution but it cannot represent the growth-decay of precipitation and non-stationary advection vector fields.In this study, we propose some nowcasting rainfall models based on advection-diffusion equation with non-stationary motion vectors. The diffusion term of this equation gives to smoother rainfall predictions for lead times and increased skill scores. The motion vectors are updated in each time step by solving a system of two-dimensional (2D) Burgers’ equation. The proposed forecasting models use the following three steps. First, an initial motion vector field is approximated using the Variational Echo Tracking (VET) algorithm. Second, a forecast is obtained for each time step by solving a time-dependent advection or advection-diffusion equation. In this step, the motion vectors are updated by solving Burgers’ equation. Lastly, forecasts are evaluated with lead times from 2.5 min to 3 h, and forecasts are compared with rain rate observations for six events over a 250×250 km2 region in southeastern South Korea.To observe the effects of the diffusion term and Burgers’ equation, four variants of the proposed modeling methods are considered, depending on the equations: advection equation (Type 1), advection and Burgers’ equations (Type 2), advection-diffusion equation (Type 3), and combination of the advection–diffusion and Burgers’ equations (Type 4). The forecasts from the Type 1 method are very similar to those of MAPLE. The other models (Type 2–4) yielded clearly better skill scores and correlation on average, with up to 3 h’ lead time. Models that use Burgers’ equation (Type 2 and Type 4) give much better scores than other methods using fixed motion vectors when the temporal variation of the motion vectors is large.

A time-space flux-corrected transport finite element formulation for solving multi-dimensional advection-diffusion-reaction equations

We present a time-space flux-corrected transport (FCT) finite element formulation for the multi-dimensional time-dependent advection-diffusion-reaction equation. Monotonic solutions can be achieved with the presented method while large time steps (with Courant number Cr>1) are used. Numerical verification, as well as a grid convergence analysis, are carried out for 1D and 2D benchmark problems. A 1D Burgers' equation with various Reynolds numbers is solved with the time-space FCT method. In addition, a 2D transport problem with (and without) a nonlinear reaction inside of a cavity is considered. Finally, the newly developed time-space FCT formulation is applied for modeling biofilm growth problems based on a continuum mathematical model. It turns out that the time-space FCT method helps to reduce numerical dissipation and guarantees comparable numerical dispersion of the solution at the same time comparing to numerical solutions obtained by a time-space finite incremental calculus (FIC) method.

Koopman operator-based model reduction for switched-system control of PDEs

We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. Using a recent convergence result for the numerical approximation via Extended Dynamic Mode Decomposition (EDMD), we show that the value of the K-ROM based objective function converges in measure to the value of the full objective function. To illustrate the results, we consider the 1D Burgers equation and the 2D Navier–Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.

Solving one- and two-dimensional unsteady Burgers' equation using fully implicit finite difference schemes

This paper introduces new fully implicit numerical schemes for solving 1D and 2D unsteady Burgers' equation. The non-linear Burgers' equation is discretized in the spatial direction by using second order Finite difference method which converts the Burgers' equation to non-linear system of ODEs. Then, the backward differentiation formula of order two (BDF-2) is employed to march the solution in the time direction. The non-linear term in the obtained system is linearized without any transformation; and it forms a system of linear algebraic equations that is solved by using Thomas's algorithm. Accuracy and performance of the proposed schemes are studied by using four test problems with Dirichlet and Neumann boundary conditions. Comparing the numerical results with exact solutions and the solutions of other schemes shows that the proposed schemes are simple, efficient and accurate even for the cases with high Reynolds numbers.

Implementation and Assessment of a Residual-Based r-Adaptation Technique on Structured Meshes

In this study, an r-adaptation technique for mesh adaptation is employed for reducing the solution discretization error, which is the error introduced due to spatial and temporal discretization of the continuous governing equations in numerical simulations. In r-adaptation, mesh modification is achieved by relocating the mesh nodes from one region to another without introducing additional nodes. Truncation error (TE) or the discrete residual is the difference between the continuous and discrete form of the governing equations. Based upon the knowledge that the discrete residual acts as the source of the discretization error in the domain, this study uses discrete residual as the adaptation driver. The r-adaptation technique employed here uses structured meshes and is verified using a series of one-dimensional (1D) and two-dimensional (2D) benchmark problems for which exact solutions are readily available. These benchmark problems include 1D Burgers equation, quasi-1D nozzle flow, 2D compression/expansion turns, and 2D incompressible flow past a Karman–Trefftz airfoil. The effectiveness of the proposed technique is evident for these problems where approximately an order of magnitude reduction in discretization error (when compared with uniform mesh results) is achieved. For all problems, mesh modification is compared using different schemes from literature including an adaptive Poisson grid generator (APGG), a variational grid generator (VGG), a scheme based on a center of mass (COM) analogy, and a scheme based on deforming maps. In addition, several challenges in applying the proposed technique to real-world problems are outlined.

Stable explicit stepwise marching scheme in ill-posed time-reversed 2D Burgers' equation

ABSTRACTThis paper constructs an unconditionally stable explicit difference scheme, marching backward in time, that can solve a limited, but important class of time-reversed 2D Burgers' initial value problems. Stability is achieved by applying a compensating smoothing operator at each time step to quench the instability. This leads to a distortion away from the true solution. However, in many interesting cases, the cumulative error is sufficiently small to allow for useful results. Effective smoothing operators based on , with real p>2, can be efficiently synthesized using FFT algorithms, and this may be feasible even in non-rectangular regions. Similar stabilizing techniques were successfully applied in other ill-posed evolution equations. The analysis of numerical stability is restricted to a related linear problem. However, extensive numerical experiments indicate that such linear stability results remain valid when the explicit scheme is applied to a significant class of time-reversed nonlinear 2D Burgers' initial value problems. As illustrative examples, the paper uses fictitiously blurred pixel images, obtained by using sharp images as initial values in well-posed, forward 2D Burgers' equations. Such images are associated with highly irregular underlying intensity data that can seriously challenge ill-posed reconstruction procedures. The stabilized explicit scheme, applied to the time-reversed 2D Burgers' equation, is then used to deblur these images. Examples involving simpler data are also studied. Successful recovery from severely distorted data is shown to be possible, even at high Reynolds numbers.

Analytical and numerical solutions for the nonlinear Burgers and advection–diffusion equations by using a semi-analytical iterative method

In this paper, exact solutions have been obtained for 1D, 2D and 3D nonlinear Burgers’ equations and systems of equations by implementing an accurate semi-analytical method. This method, originally proposed by Temimi and Ansari and herein named TAM, proved to be efficient and reliable for solving different types of linear and nonlinear problems. This method is characterized by not requiring any restrictive assumptions for the nonlinear terms. The convergence of the method is successfully presented and mathematically proved. In addition, the advection–diffusion equation is also solved by using the TAM to demonstrate the efficiency of this method. Several examples are solved either analytically or numerically, where the accuracy of the numerical solution has been demonstrated by evaluating the absolute and relative errors to show the accuracy of the proposed method. The software used in the current work is Mathematica®10.

An analytical framework of 2D diffusion, wave-like, telegraph, and Burgers’ models with twofold Caputo derivatives ordering

The purpose of the current work is to provide an analytical solution framework based on extended fractional power series expansion to solve 2D temporal–spatial fractional differential equations. For this purpose, we first present a new trivariate expansion endowed with twofold Caputo-fractional derivatives ordering, namely $$\alpha ,\,\beta \in (0,1]$$ , to study the combined effect of fractional derivatives on both temporal and spatial coordinates. Then, by virtue of this expansion, a parallel scheme of the Taylor power series solution method is utilized to extract both closed-form and supportive approximate series solutions of 2D temporal–spatial fractional diffusion, wave-like, telegraph, and Burgers’ models. The obtained closed-form solutions are found to be in harmony with the exact solutions exist in the literature when $$\alpha =\beta =1$$ , which exhibits the legitimacy and the validity of the proposed method. Moreover, the accuracy of the approximate series solutions is validated using graphical and tabular tools. Finally, a version of Taylor’s Theorem that associated with our proposed expansion is derived in terms of mixed fractional derivatives.

Freezing similarity solutions in the multi-dimensional Burgers equation

The topic of this paper is similarity solutions occurring in the multi-dimensional Burgers equation and their numerical approximation. We present a simple derivation of symmetries appearing in a family of generalizations of Burgers’ equation in d-space dimensions. We use these symmetries to obtain an equivalent partial differential algebraic equation (freezing system) that allows us to do direct long-time simulations on a fixed computational domain.As concrete examples, we are able with this method to numerically find similarity solutions for the 2D Burgers equation and observe a metastable behavior of 2D N-wave-like patterns.

Incremental proper orthogonal decomposition for PDE simulation data

We propose an incremental algorithm to compute the proper orthogonal decomposition (POD) of simulation data for a partial differential equation. Specifically, we modify an incremental matrix SVD algorithm of Brand to accommodate data arising from Galerkin-type simulation methods for time dependent PDEs. The algorithm is applicable to data generated by many numerical methods for PDEs, including finite element and discontinuous Galerkin methods. The algorithm initializes and efficiently updates the dominant POD eigenvalues and modes during the time stepping in a PDE solver without storing the simulation data. We prove that the algorithm without truncation updates the POD exactly. We demonstrate the effectiveness of the algorithm using finite element computations for a 1D Burgers’ equation and a 2D Navier–Stokes problem.

A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler–Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler–Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter $$\alpha >0$$ . Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly, namely by simulating the better-behaved 3D Euler–Voigt equations. The new criteria are only known to be sufficient criterion for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well known to occur.

Symmetry analysis of modified 2D Burgers vortex equation for unsteady case

In this paper, a symmetry analysis of the modified 2D Burgers vortex equation with a flow parameter is presented. A general form of classical and non-classical symmetries of the equation is derived. These are fundamental tools for obtaining exact solutions to the equation. In several physical cases of the parameter, the specific classical and non-classical symmetries of the equation are then obtained. In addition to rediscovering the existing solutions given by different methods, some new exact solutions are obtained with the symmetry method, showing that the symmetry method is powerful and more general for solving partial differential equations (PDEs).

Numerical discrepancy between serial and MPI parallel computations

Numerical simulations of 1D Burgers equation and 2D sloshing problem were carried out to study numerical discrepancy between serial and parallel computations. The numerical domain was decomposed into 2 and 4 subdomains for parallel computations with message passing interface. The numerical solution of Burgers equation disclosed that fully explicit boundary conditions used on subdomains of parallel computation was responsible for the numerical discrepancy of transient solution between serial and parallel computations. Two dimensional sloshing problems in a rectangular domain were solved using OpenFOAM. After a lapse of initial transient time sloshing patterns of water were significantly different in serial and parallel computations although the same numerical conditions were given. Based on the histograms of pressure measured at two points near the wall the statistical characteristics of numerical solution was not affected by the number of subdomains as much as the transient solution was dependent on the number of subdomains.

Optimal solution error quantification in variational data assimilation involving imperfect models

SummaryThe problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition. If the model is ‘perfect,’ the optimal solution (analysis) error rises because of the presence of the input data errors (background and observation errors). Then, this error is quantified by the covariance matrix, which can be approximated by the inverse Hessian of an auxiliary control problem. If the model is not perfect, the optimal solution error includes an additional component because of the presence of the model error. In this paper, we study the influence of the model error on the optimal solution error covariance, considering strong and weak constraint data assimilation approaches. For the latter, an additional equation describing the model error dynamics is involved. Numerical experiments for the 1D Burgers equation illustrate the presented theory. Copyright © 2016 John Wiley & Sons, Ltd.

Efficient approximation of Sparse Jacobians for time‐implicit reduced order models

SummaryThis paper introduces a sparse matrix discrete interpolation method to effectively compute matrix approximations in the reduced order modeling framework. The sparse algorithm developed herein relies on the discrete empirical interpolation method and uses only samples of the nonzero entries of the matrix series. The proposed approach can approximate very large matrices, unlike the current matrix discrete empirical interpolation method, which is limited by its large computational memory requirements. The empirical interpolation indices obtained by the sparse algorithm slightly differ from the ones computed by the matrix discrete empirical interpolation method as a consequence of the singular vectors round‐off errors introduced by the economy or full singular value decomposition (SVD) algorithms when applied to the full matrix snapshots. When appropriately padded with zeros, the economy SVD factorization of the nonzero elements of the snapshots matrix is a valid economy SVD for the full snapshots matrix. Numerical experiments are performed with the 1D Burgers and 2D shallow water equations test problems where the quadratic reduced nonlinearities are computed via tensorial calculus. The sparse matrix approximation strategy is compared against five existing methods for computing reduced Jacobians: (i) matrix discrete empirical interpolation method, (ii) discrete empirical interpolation method, (iii) tensorial calculus, (iv) full Jacobian projection onto the reduced basis subspace, and (v) directional derivatives of the model along the reduced basis functions. The sparse matrix method outperforms all other algorithms. The use of traditional matrix discrete empirical interpolation method is not possible for very large dimensions because of its excessive memory requirements. Copyright © 2016 John Wiley & Sons, Ltd.

2D Burgers equation with large Reynolds number using POD/DEIM and calibration

SummaryModel order reduction of the two‐dimensional Burgers equation is investigated. The mathematical formulation of POD/discrete empirical interpolation method (DEIM)‐reduced order model (ROM) is derived based on the Galerkin projection and DEIM from the existing high fidelity‐implicit finite‐difference full model. For validation, we numerically compared the POD ROM, POD/DEIM, and the full model in two cases of Re = 100 and Re = 1000, respectively. We found that the POD/DEIM ROM leads to a speed‐up of CPU time by a factor of O(10). The computational stability of POD/DEIM ROM is maintained by means of a careful selection of POD modes and the DEIM interpolation points. The solution of POD/DEIM in the case of Re = 1000 has an accuracy with error O(10−3) versus O(10−4) in the case of Re = 100 when compared with the high fidelity model. For this turbulent flow, a closure model consisting of a Tikhonov regularization is carried out in order to recover the missing information and is developed to account for the small‐scale dissipation effect of the truncated POD modes. It is shown that the computational results of this calibrated ROM exhibit considerable agreement with the high fidelity model, which implies the efficiency of the closure model used. Copyright © 2016 John Wiley & Sons, Ltd.

A new mixed finite element method based on the Crank-Nicolson scheme for Burgers’ equation

In this paper, a new mixed finite element method is used to approximate the solution as well as the flux of the 2D Burgers’ equation. Based on this new formulation, we give the corresponding stable conforming finite element approximation for the P 0 2 − P 1 pair by using the Crank-Nicolson time-discretization scheme. Optimal error estimates are obtained. Finally, numerical experiments show the efficiency of the new mixed method and justify the theoretical results.

On a transport equation with nonlocal drift

In \cite{CordobaCordobaFontelos05}, Cordoba, Cordoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions \[ \partial_t \theta +u \; \partial_x \theta = 0, \qquad u = H \theta, \] where $H$ is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Holder regularization effects of this equation and its consequences to the equation with diffusion \[ \partial_t \theta + u \; \partial_x \theta + \Lambda^\gamma \theta = 0, \qquad u = H \theta, \] where $\Lambda = (-\Delta)^{1/2}$, and $1/2 \leq \gamma <1$. Our results also apply to the model with velocity field $u = \Lambda^s H \theta$, where $s \in (-1,1)$. We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the Holder class in $C^{(s+1)/2}$, for all positive time.