Numerical and analytic studies of generalised Burgers equations

Abstract

Numerical and Analytic Studies of Generalised Burgers Equations The Burgers equation ut+uux=?uxx,???>0u_t + u u_x = \delta u_{xx}, \; \delta > 0ut?+uux?=?uxx?,?>0 is a nonlinear partial differential equation representing nonlinear wave motion in a diffusive medium. ? is a small parameter. The Burgers equation (BE) is a simple (canonical) equation which describes a balance between nonlinear convection and (small) viscous diffusion. However, the actual mathematical models such as those representing nonlinear wave motion in acoustics, electromagnetism, etc., are not so simple because of their geometric spreading and material inhomogeneity. The BE is linearizable through Cole-Hopf transformation to the heat equation. The generalised Burgers equations (GBEs), in general, do not admit linearisation. Perturbation methods have been used extensively by several investigators for solving GBEs; but these solutions are valid only over restricted space-time regions. Asymptotic methods are very powerful tools for solving differential equations; but the solutions usually involve arbitrary constants or result in the solution of the original equations on different scales. Numerical solutions are genuine solutions which may be obtained over the entire space-time range; even these pose severe difficulties especially when the initial function is discontinuous. One of the aims of the present thesis is to devise numerical methods for solving GBEs with continuous or discontinuous initial profiles and to find analytic solutions for certain space-time domains. The other major contribution of the thesis is a uniform characterisation of the GBEs by relating them via similarity transformations to a second-order nonlinear ordinary differential equation (ODE) which we refer to as generalised Euler-Painlevé equation (GEPE). This is similar to the Painlevé property for the Korteweg-de Vries (KdV) type of nonlinear dispersive wave equations.

Chapter Highlights

Chapter I: Introductory chapter highlighting the physical significance of the BE and its generalisations and briefly enumerates some physical problems of interest. Numerical schemes for solving parabolic equations of the Burgers type are presented. Some analytic solutions of the BE with appropriate initial/boundary conditions are also discussed.

Chapter II: The non-planar Burgers equation ut+u2t=?2uxxu_t + \frac{u}{2t} = \frac{\delta}{2} u_{xx}ut?+2tu?=2??uxx? (J = 1 and 2 for cylindrical and spherical symmetries, respectively) with N-wave (continuous and discontinuous) initial condition is treated. A hybrid numerical scheme combining pseudospectral and predictor-corrector implicit finite difference methods is devised and validated. For non-planar cases, computations cover evolution from sawtooth initial profile to final decay via embryonic shock, Taylor shock, thick shock, and old age. Analytic similarity forms and Reynolds number determination are discussed.

Chapter III: Evolution and decay of cylindrical and spherical nonlinear acoustic waves generated by a sinusoidal source is studied numerically. Governing equation involves ut+g(R)ux=?g(R)uxxu_t + g(R) u_x = \delta g(R) u_{xx}ut?+g(R)ux?=?g(R)uxx? with g(R)=1,12R+R2,exp?(R/R0)g(R) = 1, \frac{1}{2}R + R^2, \exp(R/R_0)g(R)=1,21?R+R2,exp(R/R0?) for plane, cylindrical, and spherical symmetries. Numerical solutions agree with asymptotic results of Scott (1981) and Enflo (1985). Saturation phenomenon and symmetry re-emergence in old age are demonstrated.

Chapters IV–VII: Investigation of several GBEs via similarity transformations leading to characterisation by GEPE: y??+ay?+f(x)yy?+g(x)y2+by?+c=0y'' + a y' + f(x) y y' + g(x) y^2 + b y' + c = 0y??+ay?+f(x)yy?+g(x)y2+by?+c=0 Examples include:

Chapter IV: GBE with general nonlinear term ut+upux=?uxx,??p>0u_t + u^p u_x = \delta u_{xx}, \; p > 0ut?+upux?=?uxx?,p>0 Self-similar solutions and bifurcation behaviour analysed. Chapter V: GBE with nonlinear damping and convection ut+upux+muq=?2uxx,??p,q>0u_t + u^p u_x + m u^q = \frac{\delta}{2} u_{xx}, \; p,q > 0ut?+upux?+muq=2??uxx?,p,q>0 Numerical and analytic results for positive and negative damping cases.Chapter VI We present the study of the GBE ut+Ju2t=p>0u_t + \frac{J u}{2 t} = p > 0ut?+2tJu?=p>0 with a general nonlinear term. For p=1J+1,J=1,2p = \frac{1}{J+1}, J = 1, 2p=J+11?,J=1,2, we are able to get an exact one-parameter family of solutions in terms of exponential and error functions. These solutions correspond to the single-hump solution of the BE. The ranges of the parameter ppp for which solutions of a connection problem to the self-similar equation exist are obtained numerically and confirmed via some integral relations. In the permissible range of ppp, the transition of (discontinuous) initial condition to self-similar form has been obtained and shown graphically.

Chapter VII We have studied the GBE ut+up=?2(1+t),??p>0u_t + u^p = \frac{\delta}{2}(1+t), \; p > 0ut?+up=2??(1+t),p>0 with a variable coefficient of viscosity. Here also we are able to obtain a single-parameter family of exact single-hump solutions for p=p?=1?n1+n.p = p^* = \frac{1-n}{1+n}.p=p?=1+n1?n?. It is found that the ODE which arises directly from the GBE through similarity transformation has solutions which either decay or oscillate at +?+\infty+? only when ?1<n<1-1 < n < 1?1<n<1. The solutions are shock-like when n=0n = 0n=0. On the other hand, they oscillate over the real line when n=?1n = -1n=?1. The solutions monotonically decay both at +?+\infty+? and ??-\infty??, that is, they have a single-hump form if p>p?p > p^*p>p?. For p<p?p < p^*p<p?, the solutions have an oscillatory behaviour either at +?+\infty+? or at ??-\infty??, or both at +?+\infty+? and ??-\infty??. For p=p?p = p^*p=p?, there exists a single-parameter family of exact single-hump solutions. Thus, the parametric value p=p?p = p^*p=p? seems to bifurcate the families of solutions which remain bounded at +?+\infty+?. In this chapter, we also present studies on two other GBEs—the inhomogeneous Burgers equation and a GBE with a variable coefficient of viscosity depending exponentially on time. While both these equations can be reduced to the Euler-Painlevé form, physically relevant solutions exist only for the inhomogeneous GBE. These solutions represent saw-tooth form. Finally, a list of all the differential equations which are either special cases of the GEPE directly or are special cases of a slightly generalised form of the GEPE in which the coefficients are made to vary with the independent variable is appended in this chapter. This brings the GEPE in contact with a much larger class of differential equations (DEs) which have appeared in diverse applications, having no direct link with GBEs. This large list covers some 65 DEs listed in the compendia of Kamke (1943) and Murphy (1960).