Nonlinear Burgers Research Articles

A Nonlinear Creep Damage Model Considering the Effect of Dry-Wet Cycles of Rocks on Reservoir Bank Slopes

Water has a crucial effect on the time-dependent behavior of rocks. The long-term cyclical fluctuations of reservoir water level lead to dry–wet (DW) cycles of rocks on reservoir bank slopes, making this influential factor more complex. To deeply understand the time-dependent behavior of rocks under DW cycles, argillite from the reservoir bank slope of Longtan Hydropower Station was used to perform a series of triaxial creep tests. Subsequently, based on analysis of creep test results after different DW cycles, a damage nonlinear Burgers viscoelastic-plastic (DNBVP) model considering the effect of saturation–dehydration cycles was proposed by introducing a nonlinear viscoplastic body and a damage variable describing DW cycles. Then, the three-dimensional creep equations of the new model were derived and its creep parameters were identified. Comparison between the theoretical curves and the test results shows that the theoretical curves of the DNBVP model were able to describe rock creep tests results after different DW cycles. Furthermore, by comparing classical creep models with the proposed model, it was found that the DNBVP model can accurately reflect the nonlinear characteristics of rocks at the accelerated creep stage. Finally, the sensitivity of the DNBVP model was analyzed and discussed, and three-dimensional central difference expressions necessary for secondary development of the new model were also derived in detail. The proposed new model with secondary development may provide a basis for improving the geotechnical design of reservoir bank slopes and the control of reservoir bank landslides.

Coupled Drift Ion Acoustic Shock waves with trapped electrons in Quantum Magnetoplasma

Coupled drift ion acoustic shock waves are studied in inhomogeneous, dissipative quantum magnetoplasmas in the presence of adiabatically trapped electrons. We have derived a nonlinear Burgers like evolution equation having a fractional nonlinearity. The quantum magnetohydrodynamics model (QMHD) has been employed to derive this nonlinear equation (2 + 1) (two spatial and one temporal) dimensions. Nonlinear analysis of the equation is carried out to show that it admits shock solutions. These shock structures are numerically investigated by using parameters of neutron stars where quantum effects are expected to dominate the dynamics of the system. These results show significant dependence on the physical parameters namely inhomogeneity, number density, ambient magnetic field, collisional frequency and the angle of propagation.

Adaptive Isostable Reduction of Nonlinear PDEs With Time Varying Parameters

Isostable reduction is a powerful technique for characterizing the transient behavior of a weakly forced, nonlinear dynamical systems in relation to a stable attractor. Practically, this reduction strategy requires small magnitude inputs so that the state remains close to the underlying attractor; when inputs become too large the reduction becomes unusable. Here, we develop an adaptive isostable coordinate framework that is valid for a continuous family of system parameters. Relations are derived that capture changes to the isostable coordinates in response to parameter changes. This information is subsequently used to define a reduction strategy valid for large magnitude but slowly varying inputs. The proposed reduction framework is compared to well-established linear and nonlinear proper orthogonal decomposition (POD) reduction techniques for simulations of the 1-dimensional nonlinear Burgers' equation with time-varying Dirichlet boundary conditions. In numerical simulations the proposed reduction strategy only requires a single mode to accurately capture system behavior. By contrast, the linear POD reduction performs poorly while the nonlinear POD strategy requires several modes to achieve comparable performance.

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.

LHAM Approach to Fractional Order Rosenau-Hyman and Burgers' Equations

Fractional calculus has been found to be a great asset in finding fractional dimension in chaos theory, in viscoelasticity diffusion, in random optimal search etc. Various techniques have been proposed to solve differential equations of fractional order. In this paper, the Laplace-Homotopy Analysis Method (LHAM) is applied to obtain approximate analytic solutions of the nonlinear Rosenau-Hyman Korteweg-de Vries (KdV), K(2, 2), and Burgers' equations of fractional order with initial conditions. The solutions of these equations are calculated in the form of convergent series. The solutions obtained converge to the exact solution when α = 1, showing the reliability of LHAM.

Effective Method for Solving Different Types of Nonlinear Fractional Burgers’ Equations

In this study, a relatively new method to solve partial differential equations (PDEs) called the fractional reduced differential transform method (FRDTM) is used. The implementation of the method is based on an iterative scheme in series form. We test the proposed method to solve nonlinear fractional Burgers equations in one, two coupled, and three dimensions. To show the efficiency and accuracy of this method, we compare the results with the exact solutions, as well as some established methods. Approximate solutions for different values of fractional derivatives together with exact solutions and absolute errors are represented graphically in two and three dimensions. From all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional partial differential equations over existing methods.

Solving time fractional Burgers\u2019 and Fisher\u2019s equations using cubic B-spline approximation method

This article presents a numerical algorithm for solving time fractional Burgers’ and Fisher’s equations using cubic B-spline finite element method. The L1 formula with Caputo derivative is used to discretized the time fractional derivative, whereas the Crank–Nicolson scheme based on cubic B-spline functions is used to interpolate the solution curve along the spatial grid. The numerical scheme has been implemented on three test problems. The obtained results indicate that the proposed method is a good option for solving nonlinear fractional Burgers’ and Fisher’s equations. The error norms L_{2} and L_{infty } have been calculated to validate the efficiency and accuracy of the presented algorithm.

A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations

PurposeThis paper aims to develop a new high accuracy numerical method based on off-step non-polynomial spline in tension approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations on a graded mesh. The spline method reported here is third order accurate in space and second order accurate in time. The proposed spline method involves only two off-step points and a central point on a graded mesh. The method is two-level implicit in nature and directly derived from the continuity condition of the first order space derivative of the non-polynomial tension spline function. The linear stability analysis of the proposed method has been examined and it is shown that the proposed two-level method is unconditionally stable for a linear model problem. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, the authors have solved the generalized Burgers-Huxley equation, generalized Burgers-Fisher equation, coupled Burgers-equations and parabolic equation in polar coordinates. The authors show that the proposed method enables us to obtain the high accurate solution for high Reynolds number.Design/methodology/approachIn this method, the authors use only two-level in time-direction, and at each time-level, the authors use three grid points for the unknown function u(x,t) and two off-step points for the known variable x in spatial direction. The methodology followed in this paper is the construction of a non-polynomial spline function and using its continuity properties to obtain consistency condition, which is third order accurate on a graded mesh and fourth order accurate on a uniform mesh. From this consistency condition, the authors derive the proposed numerical method. The proposed method, when applied to a linear equation is shown to be unconditionally stable. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the proposed method.FindingsThe paper provides a third order numerical scheme on a graded mesh and fourth order spline method on a uniform mesh obtained directly from the consistency condition. In earlier methods, consistency conditions were only second order accurate. This brings an edge over other past methods. Also, the method is directly applicable to physical problems involving singular coefficients. So no modification in the method is required at singular points. This saves CPU time and computational costs.Research limitations/implicationsThere are no limitations. Obtaining a high accuracy spline method directly from the consistency condition is a new work. Also being an implicit method, this method is unconditionally stable.Practical implicationsPhysical problems with singular and non-singular coefficients are directly solved by this method.Originality/valueThe paper develops a new method based on non-polynomial spline approximations of order two in time and three (four) in space, which is original and has lot of value because many benchmark problems of physical significance are solved in this method.

Modeling Asymmetric Interactions in Economy

We consider a general nonlinear kinetic type equation that can describe the time evolution of a variable related to an economical state of an individual agent of the system. We assume asymmetric interactions between the agents. We show that in a corresponding limit, it is asymptotically equivalent to a nonlinear inviscid Burgers type equation.

Numerical Solution of Burger's Type Equation Using Finite Element Collocation method with Strang Splitting

The nonlinear Burgers equation, which has a convection term, a viscosity term and a time dependent term in its structure, has been splitted according to the time term and then has been solved by finite element collocation method using cubic B-spline bases. By splitting the equation U_{t}+UU_{x}=vU_{xx} two simpler sub problems U_{t}+UU_{x}=0 and U_{t}-vU_{xx}=0 have been obtained. A discretization process has been performed for each of these sub-problems and the stability analyzes have been carried out by Fourier (von Neumann) series method. Then, both sub-problems have been solved using the Strang splitting technique to obtain numerical results. To see the effectiveness of the present method, which is a combination of finite element method and Strang splitting technique, we have calculated the frequently used error norms ‖e‖₁, L₂ and L_{∞} in the literature and have made a comparison between exact and a numerical solution.

Improved nonlinear Burgers shear creep model based on the time-dependent shear strength for rock

Shear strength is an important mechanical index of rock. The adjustment and reorganization of rock structure can be reflected by the variation of shear strength. During rock rheology, the shear strength decreases with time due to rock damage. Therefore, from the point of difference of shear strength, the mechanical properties of rock can be effectively studied. In this paper, for the state of direct shear test, the Kachanov creep damage law was adopted to describe the time characteristics of the rock shear strength during the accelerated creep stage. A nonlinear viscoplastic element on the basis of time-dependent shear strength was established by connecting the plastic element representing shear strength with the viscous element in parallel. After introducing the nonlinear viscoplastic element into the classic Burgers model, the rationality of the model was verified by the shear creep test results of rock discontinuity. Results showed that the modified Burgers model can reflect the mechanical properties of rock in three creep stages. In addition, the model parameter δ can also reflect the evolution of internal cracks in rock during creep.

Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations

This article proposes adaptive iterative splitting methods to solve Multiphysics problems, which are related to convection–diffusion–reaction equations. The splitting techniques are based on iterative splitting approaches with adaptive ideas. Based on shifting the time-steps with additional adaptive time-ranges, we could embedded the adaptive techniques into the splitting approach. The numerical analysis of the adapted iterative splitting schemes is considered and we develop the underlying error estimates for the application of the adaptive schemes. The performance of the method with respect to the accuracy and the acceleration is evaluated in different numerical experiments. We test the benefits of the adaptive splitting approach on highly nonlinear Burgers’ and Maxwell–Stefan diffusion equations.

A spatial sixth-order CCD-TVD method for solving multidimensional coupled Burgers’ equation

In this paper, a high-order compact difference scheme is proposed for solving multidimensional nonlinear Burgers’ equation. The three-stage third-order total variation diminishing (TVD) Runge–Kutta scheme is employed in time, and the three-point combined compact difference (CCD) scheme is used for spatial discretization. The proposed TVD-CCD method is free of using Hopf–Cole transformation, and treats the nonlinear term explicitly. Thus it is very efficient and easy to implement. Our method is effective to capture shock wave, third-order accurate in time, and sixth-order accurate in space. In addition, we show the unique solvability of the CCD system under non-periodic boundary conditions. Numerical experiments are given to demonstrate the high efficiency and accuracy of the proposed method.

A fourth-order B-spline collocation method for nonlinear Burgers\u2013Fisher equation

A fourth-order B-spline collocation method has been applied for numerical study of Burgers–Fisher equation, which illustrates many situations occurring in various fields of science and engineering including nonlinear optics,gas dynamics, chemical physics, heat conduction, and so on. The present method is successfully applied to solve the Burgers–Fisher equation taking into consideration various parametric values. The scheme is found to be convergent. Crank–Nicolson scheme has been employed for the discretization. Quasi-linearization technique has been employed to deal with the nonlinearity of equations. The stability of the method has been discussed using Fourier series analysis (von Neumann method), and it has been observed that the method is unconditionally stable. In order to demonstrate the effectiveness of the scheme, numerical experiments have been performed on various examples. The solutions obtained are compared with results available in the literature, which shows that the proposed scheme is satisfactorily accurate and suitable for solving such problems with minimal computational efforts.

Three dimensional nonlinear shock waves in inhomogeneous plasmas with different size dust grains and external magnetized field

The evolution of three dimensional nonlinear shock waves is investigated in a dusty plasma with inhomogeneous particles’ density, nonadiabatic dust charge variation, external magnetized field and power law dust size distribution. For this purpose, a modified nonlinear Korteweg-de Vries Burgers equation containing variable coefficients is obtained by reductive perturbation method. The effects of inhomogeneity, dust size distribution, external magnetized field, dust charge variation and obliqueness parameter on shock structures are numerically examined in great detail. Furthermore, research results show that oscillatory shock waves and monotone shock waves exist and transform each other in this system.

Adaptive activation functions accelerate convergence in deep and physics-informed neural networks

We employ adaptive activation functions for regression in deep and physics-informed neural networks (PINNs) to approximate smooth and discontinuous functions as well as solutions of linear and nonlinear partial differential equations. In particular, we solve the nonlinear Klein-Gordon equation, which has smooth solutions, the nonlinear Burgers equation, which can admit high gradient solutions, and the Helmholtz equation. We introduce a scalable hyper-parameter in the activation function, which can be optimized to achieve best performance of the network as it changes dynamically the topology of the loss function involved in the optimization process. The adaptive activation function has better learning capabilities than the traditional one (fixed activation) as it improves greatly the convergence rate, especially at early training, as well as the solution accuracy. To better understand the learning process, we plot the neural network solution in the frequency domain to examine how the network captures successively different frequency bands present in the solution. We consider both forward problems, where the approximate solutions are obtained, as well as inverse problems, where parameters involved in the governing equation are identified. Our simulation results show that the proposed method is a very simple and effective approach to increase the efficiency, robustness and accuracy of the neural network approximation of nonlinear functions as well as solutions of partial differential equations, especially for forward problems. We theoretically prove that in the proposed method, gradient descent algorithms are not attracted to suboptimal critical points or local minima. Furthermore, the proposed adaptive activation functions are shown to accelerate the minimization process of the loss values in standard deep learning benchmarks using CIFAR-10, CIFAR-100, SVHN, MNIST, KMNIST, Fashion-MNIST, and Semeion datasets with and without data augmentation.

A new fractal nonlinear Burgers' equation arising in the acoustic signals propagation

In this letter, we consider the new nonlinear Burgers' equation engaging local fractional derivative for the first time. With the use of the travelling‐wave transformation of non‐differentiable type, some exact solutions for the new nonlinear Burgers' equation are discussed in detail. The obtained results are accurate and efficient for the descriptions of the acoustic signals propagation in the fractal stratified media.

A simple algorithm for numerical solution of nonlinear parabolic partial differential equations

In this paper, numerical solution of nonlinear two-dimensional parabolic partial differential equations with initial and Dirichlet boundary conditions is considered. The time derivative is approximated using finite difference scheme whereas space derivatives are approximated using Haar wavelet collocation method. The proposed method is developed for semilinear and quasilinear cases, however, it can easily be extended to other types of nonlinearities as well. The proposed method is also illustrated for nonlinear heat equation and Burgers’ equation. The proposed method is implemented upon five test problems and the numerical results are shown using tables and figures. The numerical results validate the accuracy and efficiency of the proposed method.

Nonlinear viscoelastic-plastic material modelling for the behaviour of ice in ice-structure interactions

In this paper, a three-dimensional constitutive ice model with nonlinear viscoelastic and plastic components acting in series is proposed for the dynamic simulation of ice-structure interactions. The former component corresponds to a nonlinear Burgers model, in which the deviatoric behaviour is viscoelastic and the volumetric behaviour is elastic. The Burgers model is combined with the well accepted Glen's law and Ashby and Duval’ law to describe the time-dependent behaviour and distribution in relaxation time. A hydrostatic pressure- and octahedral shear stress-dependent Tsai-Wu-type yield criterion is adopted to invoke the plastic state. The proposed ice model is assumed to be isotropic and is implemented in the commercial software LS-DYNA as a user-defined material. Verification of this model is implemented via simulations of creep and indentation experiments and via the simulation of a scenario of ice-rigid steel plate collision. The simulation of creep experiments shows that this material model can reflect the viscoelastic properties of ice quite well. The calculated contact force and pressure-area curves agree well with the results of indentation experiments. And the ice-rigid steel plate collision example yields reasonable results, indicating that the proposed model has a good capacity for describing the brittle behaviour of ice. The numerical verification proves that the proposed ice material model has a wide range of potential applications.

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.