Two-dimensional Burgers Research Articles

A Cell-Centered Semi-Lagrangian Finite Volume Method for Solving Two-Dimensional Coupled Burgers’ Equations

A cell-centered finite volume semi-Lagrangian method is presented for the numerical solution of two-dimensional coupled Burgers’ problems on unstructured triangular meshes. The method combines a modified method of characteristics for the time integration and a cell-centered finite volume for the space discretization. The new method belongs to fractional-step algorithms for which the convection and the viscous parts in the coupled Burgers’ problems are treated separately. The crucial step of interpolation in the convection step is performed using two local procedures accounting for the element where the departure point is located. The resulting semidiscretized system is then solved using a third-order explicit Runge-Kutta scheme. In contrast to the Eulerian-based methods, we apply the new method for each time step along the characteristic curves instead of the time direction. The performance of the current method is verified using different examples for coupled Burgers’ problems with known analytical solutions. We also apply the method for simulation of an example of coupled Burgers’ flows in a complex geometry. In these test problems, the new cell-centered finite volume semi-Lagrangian method demonstrates its ability to accurately resolve the two-dimensional coupled Burgers’ problems.

A Triangular Spectral Element Method for the 2-D Viscous Burgers Equation

A triangular spectral element method is established for the two-dimensional viscous Burgers equation. In the spatial direction, a new type of mapping is applied. We splice the local basis functions on each triangle into a global basis function. The second-order Crank-Nicolson/ leap-frog (CNLF) method is used for discretization in the time direction. Due to the use of a quasi-interpolation operator, the nonlinear term can be handled conveniently. We give the fully discrete scheme of the method and the implementation process of the algorithm. Numerical examples verify the effectiveness of this method.

A Compact Subcell WENO Limiting Strategy Using Immediate Neighbors for Runge-Kutta Discontinuous Galerkin Methods for Unstructured Meshes

In this paper, we generalize the compact subcell weighted essentially non oscillatory (CSWENO) limiting strategy for Runge-Kutta discontinuous Galerkin method developed recently in [1] for structured meshes to unstructured triangular meshes. The main idea of the limiting strategy is to divide the immediate neighbors of a given cell as subcells into the required stencil and to use a WENO reconstruction for limiting. This strategy can be applied for any type of WENO reconstruction. We have used the WENO reconstruction proposed in [2] and provided accuracy tests and results for two-dimensional Burgers’ equation and two dimensional Euler equations to illustrate the performance of this limiting strategy.

Numerical solution of one- and two-dimensional time-fractional Burgers equation via Lucas polynomials coupled with Finite difference method

In this article, a numerical technique based on polynomials is proposed for the solution of one and two-dimensional time-fractional Burgers equation. First, the given problem is reduced to time discrete form using θ-weighted scheme. Then, with the help of Lucas and Fibonacci polynomials the given PDEs transformed to system of algebraic equations which is easy to solve. The proposed algorithm is validated by solving some numerical examples. Despite this, convergence analysis of the scheme is briefly discussed and verified numerically. The main objective of this paper is to show that polynomial based method is convenient for 1D and 2D nonlinear time-fractional partial differential equations (TFPDEs). Efficiency and performance of the proposed technique are examined by calculating L2 and L∞ error norms. Obtained accurate results confirm applicability and efficiency of the method.

Two space–time methods for solving Burgers’ equation

Burgers’ equation is a kind of quasi-linear important partial differential equation appeared in many fields. The nonlinear term contained in this equation results in difficulty in finding its high precision numerical solution. Most existed research focused on one-dimensional and two-dimensional Burgers’ systems. Seldom research studied the three-dimensional Burgers’ problem. In this paper, two space–time methods based on the polynomial particular solutions (ST-MPPS) and the polynomial used as basis functions directly, have been proposed to solve three-dimensional Burgers’ equation at different final time with different Reynolds numbers. The numerical examples show that the greatest attraction of the ST-MPPS is its high precision and robustness. When using the space–time polynomial functions method, some interesting phenomenon appears with the Reynolds number increasing.

Extended symmetry analysis of two-dimensional degenerate Burgers equation

We carry out the extended symmetry analysis of a two-dimensional degenerate Burgers equation. Its complete point-symmetry group is found using the algebraic method, and all its generalized symmetries are proved equivalent to its Lie symmetries. We also prove that the space of conservation laws of this equation is infinite-dimensional and is naturally isomorphic to the solution space of the (1+1)-dimensional backward linear heat equation. Lie reductions of the two-dimensional degenerate Burgers equation are comprehensively studied in the optimal way and new Lie invariant solutions are constructed. We additionally consider solutions that also satisfy an analogous nondegenerate Burgers equation. In total, we construct four families of solutions of two-dimensional degenerate Burgers equation that are expressed in terms of arbitrary (nonzero) solutions of the (1+1)-dimensional linear heat equation. Various kinds of hidden symmetries and hidden conservation laws (local and potential ones) are discussed as well. As a by-product, we exhaustively describe generalized symmetries, cosymmetries and conservation laws of the transport equation, also called the inviscid Burgers equation, and construct new invariant solutions of the nonlinear diffusion and diffusion–convection equations with power nonlinearities of degree −1/2.

A Composite Algorithm for Numerical Solutions of Two-Dimensional Coupled Burgers’ Equations

In this study, a new composite algorithm with the help of the finite difference and the modified cubic trigonometric B-spline differential quadrature method is developed. The developed method was applied to two-dimensional coupled Burgers’ equation with initial and Dirichlet boundary conditions for computational modeling. The established algorithm is better than the traditional differential quadrature algorithm proposed in literature due to more smoothness of cubic trigonometric B-spline functions. In the development of the algorithm, the first step is semidiscretization in time with the forward finite difference method. Furthermore, the obtained system is fully discretized by the modified cubic trigonometric B-spline differential quadrature method. Finally, we obtain coupled Lyapunov systems of linear equations, which are analyzed by the MATLAB solver for the system. Moreover, comparative study of these solutions with the numerical and exact solutions which are appeared in the literature is also discussed. Finally, it is found that there is good suitability between exact solutions and numerical solutions obtained by the developed composite algorithm. The technique can be extended for various multidimensional Burgers’ equations after some modifications.

High order semi-implicit weighted compact nonlinear scheme for viscous Burgers’ equations

This paper develops a high order semi-implicit weighted compact nonlinear scheme (WCNS) for one- and two-dimensional viscous Burgers’ equations. This semi-implicit WCNS combines a fifth-order WCNS with a third-order implicit-explicit Runge–Kutta (IMEX-RK) time-stepping method. The advection terms of viscous Burgers’ equations are treated explicitly, while the viscosity terms are treated implicitly. Stability analysis shows that the CFL condition of the semi-implicit WCNS is controlled only by the advection terms. Compared to the explicit time-stepping method, the semi-implicit method has the advantage in terms of computational efficiency. Numerical results validate the accuracy and efficiency of the semi-implicit WCNS. This method can also solve viscous Burgers’ equations with large Reynolds numbers and has high-resolution shock-capturing ability.

An implicit finite difference scheme for the numerical solutions of two-dimensional Burgers equations

An implicit finite difference scheme for the numerical solutions of two-dimensional Burgers equations

二维Burgers方程的分裂高阶有限差分方法

二维Burgers方程的分裂高阶有限差分方法

Space–time generalized finite difference nonlinear model for solving unsteady Burgers’ equations

In this study, the space–time (ST) generalized finite difference method (GFDM) was combined with Newton’s method to stably and accurately solve two-dimensional unsteady Burgers’ equations. In the coupled ST approach, the time axis is selected as a spatial axis; thus, the temporal derivative in governing equations is treated as a spatial derivative. In general, the GFDM is an optimal meshless collocation method for solving partial differential equations. Moreover, one can avoid the construction of a mesh for simulation by using the GFDM. The derivatives at each node are described as a linear combination of nearby functional values by using weighting coefficients in the computational domain. Due to the property of the moving least-square approximation in the GFDM, the resultant matrix system can be formed as a sparse matrix so that the GFDM is suitable for solving large-scale problems. In this study, two benchmark examples were used to demonstrate the consistency and accuracy of the proposed ST meshless numerical scheme.

Radially symmetric stationary wave for two-dimensional Burgers equation

We are concerned with the radially symmetric stationary wave for the exterior problem of two-dimensional Burgers equation. A sufficient and necessary condition to guarantee the existence of such a stationary wave is given and it is also shown that the stationary wave satisfies nice decay estimates and is time-asymptotically nonlinear stable under radially symmetric initial perturbation.

Separation of a binary gas mixture by thermal diffusion in a two-dimensional cascade of many small cavities

A new method for separating a binary gas mixture, such as hydrogen and carbon dioxide, is proposed and investigated numerically and experimentally. As a pressure driven flow of the mixture goes through a two-dimensional Burgers cascade, the components are separated due to thermal diffusion and the amplifying effect of the cascade. Unlike traditional cascades in isotope separation, active elements like pumps are absent. The device is made thin enough so that the flow and the concentration distribution reaches an equilibrium within each cell. We have observed separation of up to about 9% in hydrogen concentration in our prototype array consisting of over 19,000 cavities placed under a 160-degree temperature difference.

Comments on the Paper “Lie Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation”

This comment is devoted to the paper “Lie Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation” (Symmery, 2020, vol.12, 170), in which several results are either incorrect, or incomplete, or misleading.

A compact subcell WENO limiting strategy using immediate neighbours for Runge-Kutta discontinuous Galerkin methods

A compact subcell weighted essentially non oscillatory (CSWENO) limiter is proposed for the solution of hyperbolic conservation laws with discontinuous Galerkin Method which uses only the immediate neighbours of a given cell. These neighbours are divided into the required stencil for WENO reconstruction and an existing WENO limiting strategy is used. Accuracy tests and results for one-dimensional and two-dimensional Burgers' equation and one-dimensional and two-dimensional Euler equations for Cartesian meshes are presented using this limiter. Comparisons with the parent WENO limiter are provided wherever appropriate and the performance of the current limiter is found to be slightly better than the parent WENO limiter for higher orders.

Lie Symmetry Analysis, Explicit Solutions and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation

In this paper, we investigate a spatially two-dimensional Burgers–Huxley equation that depicts the interaction between convection effects, diffusion transport, reaction gadget, nerve proliferation in neurophysics, as well as motion in liquid crystals. We have used the Lie symmetry method to study the vector fields, optimal systems of first order, symmetry reductions, and exact solutions. Furthermore, using the power series method, a set of series solutions are obtained. Finally, conservation laws are derived using optimal systems.

An Alternating Segment Explicit-Implicit Scheme with Intrinsic Parallelism for Burgers’ Equation

In this paper, an alternating segment explicit-implicit (ASE-I) scheme with intrinsic parallelism for Burgers’ equation is proposed. The scheme has second-order truncation error in space and it is linear stable by analysis of the linearization procedure. The ASE-I scheme can be extended to solve two-dimensional Burgers’ equations by alternating direction implicit (ADI) technique. Numerical experiments show the new scheme is efficient and reliable.

A New Compact Scheme in Exponential Form for Two-Dimensional Time-Dependent Burgers' and Navier-Stokes Equations

A New Compact Scheme in Exponential Form for Two-Dimensional Time-Dependent Burgers' and Navier-Stokes Equations

Predictor–Corrector Nodal Integral Method for simulation of high Reynolds number fluid flow using larger time steps in Burgers ’ equation

Nodal Integral Methods (NIM), although frequently used for solving neutron transport equations, have not found wider acceptance for solving fluid flow problems. One of the pivotal reasons behind this is the lack of efficient non-linear solvers for the system of coupled algebraic equations obtained by these methods, thus making such methods prohibitively expensive for highly nonlinear (higher Reynolds number) fluid flow problems. Successive improvements in earlier attempts resulted in an improved version called Modified NIM (MNIM) scheme; following which, a further modified version appeared (called as M2NIM), that used the concept of the delayed coefficients. Upon comparing the solutions obtained using MNIM and M2NIM schemes, respectively, it was observed that although the modified scheme with delayed coefficients (M2NIM) has significantly faster convergence, it is less accurate. The convergence and accuracy are very likely to suffer to a significant extent in the high Reynolds number fluid flows, especially when a large time-step is to be considered. In order to resolve these difficulties, a new type of physics-based predictor–corrector algorithm is proposed in the present study. In the proposed computational algorithm, the linearized M2NIM scheme is used to predict the solution, whereas the MNIM scheme is utilized to improve the predictive guess. The novelty of such a hybrid numerical algorithm comes from the fact that it combines the advantage of the faster convergence of M2NIM with the accuracy of the MNIM scheme. The algorithm also benefits from the unique inclusion of Jacobian-free Newton-Krylov (JFNK) method, which helps in getting rid of the formation of large Jacobian matrices, thereby reducing unnecessary computational overhead. The proposed methodology has been applied to solve a non-linear convection–diffusion problem, represented by the Burgers’ equation. The computational results for bothone-dimensional and two-dimensional Burgers’ equation are presented to demonstrate the effectiveness of the developed novel algorithm, as well as, the advantage it offers over the existing numerical 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.