1,497 research outputs found
Nonlinear hyperbolic waves in multidimensions
No full textThe propagation of curved, nonlinear wavefronts and shock fronts are very complex phenomena. Since the 1993 publication of his work Propagation of a Curved Shock and Nonlinear Ray Theory, author Phoolan Prasad and his research group have made significant advances in the underlying theory of these phenomena. This volume presents their results and provides a self-contained account and gradual development of mathematical methods for studying successive positions of these fronts.Nonlinear Hyperbolic Waves in Multidimensions includes all introductory material on nonlinear hyperbolic waves and the theory of shock waves. The author derives the ray theory for a nonlinear wavefront, discusses kink phenomena, and develops a new theory for plane and curved shock propagation. He also derives a full set of conservation laws for a front propagating in two space dimensions, and uses these laws to obtain successive positions of a front with kinks. The treatment includes examples of the theory applied to converging wavefronts in gas dynamics, a graphical presentation of the results of extensive numerical computations, and an extension of Fermat''s principle. There is also a chapter containing approximate equations used to discuss stability of steady transonic flows.Full of new and original results, Nonlinear Hyperbolic Waves in Multidimensions is your only opportunity to explore a full treatment of these recent findings in book form. The material presented in this volume will prove useful not only for solving practical problems, but also in raising many difficult but important mathematical questions that remain open
An asymptotic derivation of weakly nonlinear ray theory
No full textUsing a method of expansion similar to Chapman-Enskog expansion, a new formal perturbation scheme based on high frequency approximation has been constructed. The scheme leads to an eikonal equation in which the leading order amplitude appears. The transport equation for the amplitude has been deduced with an error O(epsilon (2)) where epsilon is the small parameter appearing in the high frequency approximation. On a length scale over which Choquet-Bruhats theory is valid, this theory reduces to the former. The theory is valid on a much larger length scale and the leading order terms give the weakly nonlinear ray theory (WNLRT) of Prasad, which has been very successful in giving physically realistic results and also in showing that the caustic of a linear theory is resolved when nonlinear effects are included. The weak shock ray theory with infinite system of compatibility conditions also follows from this theory
On shock dynamics
No full textThis is in continuation of our paper 'On the propagation of a multi-dimensional shock of arbitrary strength' published earlier in this journal (Srinivasan and Prasad ). We had shown in our paper that Whitham’s shock dynamics, based on intuitive arguments, cannot be relied on for flows other than those involving weak shocks and that too with uniform flow behind the shock. Whitham refers to this as misinterpretation of his approximation and claims that his theory is not only correct but also provides a natural closure of the open system of the equations of Maslov. The main aim of this note is to refute Whitham’s claim with the help of an example and a numerical integration of a problem in gasdynamics
An asymptotic derivation of weakly nonlinear ray theory
No full textUsing a method of expansion similar to Chapman-Enskog expansion, a new formal perturbation scheme based on high frequency approximation has been constructed. The scheme leads to an eikonal equation in which the leading order amplitude appears. The transport equation for the amplitude has been deduced with an error 0(ε2) where ε is the small parameter appearing in the high frequency approximation. On a length scale over which Choquet-Bruhat's theory is valid, this theory reduces to the former. The theory is valid on a much larger length scale and the leading order terms give the weakly nonlinear ray theory (WNLRT) of Prasad, which has been very successful in giving physically realistic results and also in showing that the caustic of a linear theory is resolved when nonlinear effects are included. The weak shock ray theory with infinite system of compatibility conditions also follows from this theory
On shock dynamics
No full textThis is in continuation of our paper On the propagation of a multi-dimensional shock of arbitrary strength’ published earlier in this journal (Srinivasan and Prasad [9]). We had shown in our paper that Whitham’s shock dynamics, based on intuitive arguments, cannot be relied on for flows other than those involving weak shocks and that too with uniform flow behind the shock. Whitham [12] refers to this as misinterpretation of his approximation and claims that his theory is not only correct but also provides a natural closure of the open system of the equations of Maslov [3]. The main aim of this note is to refute Whitham’s claim with the help of an example and a numerical integration of a problem in gasdynamics
Numerical Study Of Regularization Methods For Elliptic Cauchy Problems
No full textCauchy problems for elliptic partial differential equations arise in many important applications, such as, cardiography, nondestructive testing, heat transfer, sonic boom produced by a maneuvering aerofoil, etc. Elliptic Cauchy problems are typically ill-posed, i.e., there may not be a solution for some Cauchy data, and even if a solution exists uniquely, it may not depend continuously on the Cauchy data. The ill-posedness causes numerical instability and makes the classical numerical methods inappropriate to solve such problems. For Cauchy problems, the research on uniqueness, stability, and efficient numerical methods are of significant interest to mathematicians. The main focus of this thesis is to develop numerical techniques for elliptic Cauchy problems.
Elliptic Cauchy problems can be approached as data completion problems, i.e., from over-specified Cauchy data on an accessible part of the boundary, one can try to recover missing data on the inaccessible part of the boundary. Then, the Cauchy problems can be solved by finding a so-lution to a well-posed boundary value problem for which the recovered data constitute a boundary condition on the inaccessible part of the boundary.
In this thesis, we use natural linearization approach to transform the linear Cauchy problem into a problem of solving a linear operator equation. We consider this operator in a weaker image space H−1, which differs from the previous works where the image space of the operator is usually considered as L2 . The lower smoothness of the image space will make a problem a bit more ill-posed. But under such settings, we can prove the compactness of the considered operator. At the same time, it allows a relaxation of the assumption concerning noise.
The numerical methods that can cope with these ill-posed operator equations are the so called regularization methods. One prominent example of such regularization methods is Tikhonov regularization which is frequently used in practice. Tikhonov regularization can be considered as a least-squares tracking of data with a regularization term. In this thesis we discuss a possibility to improve the reconstruction accuracy of the Tikhonov regularization method by using an iterative modification of Tikhonov regularization. With this iterated Tikhonov regularization the effect of the penalty term fades away as iterations go on.
In the application of iterated Tikhonov regularization, we find that for severely ill-posed problems such as elliptic Cauchy problems, discretization has such a powerful influence on the accuracy of the regularized solution that only with some reasonable discretization level, desirable accuracy can be achieved. Thus, regularization by projection method which is commonly known as self-regularization is also considered in this thesis. With this method, the regularization is achieved only by discretization along with an appropriate choice of discretization level.
For all regularization methods, the choice of an appropriate regularization parameter is a crucial issue. For this purpose, we propose the balancing principle which is a recently introduced powerful technique for the choice of the regularization parameter. While applying this principle, a balance between the components related to the convergence rate and stability in the accuracy estimates has to be made. The main advantage of the balancing principle is that it can work in an adaptive way to obtain an appropriate value of the regularization parameter, and it does not use any quantitative knowledge of convergence rate or stability. The accuracy provided by this adaptive strategy is worse only by a constant factor than one could achieve in the case of known stability and convergence rates. We apply the balancing principle in both iterated Tikhonov regularization and self-regularization methods to choose the proper regularization parameters.
In the thesis, we also investigate numerical techniques based on iterative Tikhonov regular-ization for nonlinear elliptic Cauchy problems. We consider two types of problems. In the first kind, the nonlinear problem can be transformed to a linear problem while in the second kind, linearization of the nonlinear problem is not possible, and for this we propose a special iterative method which differs from methods such as Landweber iteration and Newton-type method which are usually based on the calculation of the Frech´et derivative or adjoint of the equation.
Abundant examples are presented in the thesis, which illustrate the performance of the pro-posed regularization methods as well as the balancing principle. At the same time, these examples can be viewed as a support for the theoretical results achieved in this thesis.
In the end of this thesis, we describe the sonic boom problem, where we first encountered the ill-posed nonlinear Cauchy problem. This is a very difficult problem and hence we took this problem to provide a motivation for the model problems. These model problems are discussed one by one in the thesis in the increasing order of difficulty, ending with the nonlinear problems in Chapter 5.
The main results of the dissertation are communicated in the article [35]
3-D kinematical conservation laws (KCL): Evolution of a surface in R-3 - in particular propagation of a nonlinear wavefront
No full text3-D KCL are equations of evolution of a propagating surface (or a wavefront) Omega(t), in 3-space dimensions and were first derived by Giles, Prasad and Ravindran in 1995 assuming the motion of the surface to be isotropic. Here we discuss various properties of these 3-D KCL.These are the most general equations in conservation form, governing the evolution of Omega(t) with singularities which we call kinks and which are curves across which the normal n to Omega(t) and amplitude won Omega(t) are discontinuous. From KCL we derive a system of six differential equations and show that the KCL system is equivalent to the ray equations of 2, The six independent equations and an energy transport equation (for small amplitude waves in a polytropic gas) involving an amplitude w (which is related to the normal velocity m of Omega(t)) form a completely determined system of seven equations. We have determined eigenvalues of the system by a very novel method and find that the system has two distinct nonzero eigenvalues and five zero eigenvalues and the dimension of the eigenspace associated with the multiple eigenvalue 0 is only 4. For an appropriately defined m, the two nonzero eigenvalues are real when m > 1 and pure imaginary when m < 1. Finally we give some examples of evolution of weakly nonlinear wavefronts
Propagation of multidimensional nonlinear waves and kinematical conservation laws
No full textThis book formulates the kinematical conservation laws (KCL), analyses them and presents their applications to various problems in physics. Finally, it addresses one of the most challenging problems in fluid dynamics: finding successive positions of a curved shock front. The topics discussed are the outcome of collaborative work that was carried out mainly at the Indian Institute of Science, Bengaluru, India. The theory presented in the book is supported by referring to extensive numerical results. The book is organised into ten chapters. Chapters 1–4 offer a summary of and briefly discuss the theory of hyperbolic partial differential equations and conservation laws. Formulation of equations of a weakly nonlinear wavefront and those of a shock front are briefly explained in Chapter 5, while Chapter 6 addresses KCL theory in space of arbitrary dimensions. The remaining chapters examine various analyses and applications of KCL equations ending in the ultimate goal-propagation of a three-dimensional curved shock front and formation, propagation and interaction of kink lines on it
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