Hyperbolic conservation laws form a special class of partial differential equations. They describe phenomena that involve conserved quantities and their solutions show discontinuities which reflect the formation of shock waves. We consider one-dimensional systems of hyperbolic conservation laws and produce approximations using finite difference, finite volume and finite element methods. Due to stability issues of classical finite element methods for hyperbolic conservation laws, we study the discontinuous Galerkin method, which was recently introduced. The method involves completely discontinuous basis functions across each element and it can be considered as a combination of finite volume and finite element methods. We illustrate the implementation of discontinuous Galerkin method using Legendre polynomials, in case of scalar equations and in case of quasi-linear systems, and we review important theoretical results about stability and convergence of the method. The applications of finite volume and discontinuous Galerkin methods to linear and non-linear scalar equations, as well as to the system of elastodynamics, are exhibited.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 185-191) and index.

Many applications in engineering practice can be described by thehyperbolic partial differential equations (PDEs). Numerical modeling of this typeof equations often involves large gradients or shocks, which makes it achallenging task for conventional numerical methods to accurately simulate suchsystems. Thus developing accurate and efficient shock capturing numericalschemes becomes important for the study of hyperbolic equations.In this dissertation, a detailed study of the numerical methods for linearand nonlinear unsteady hyperbolic equations was carried out. A new finitedifference shock capturing scheme of finite volume style was developed. Thisscheme is based on the high order Pad?? type compact central finite differencemethod with the weighted essentially non-oscillatory (WENO) reconstruction toeliminate non-physical oscillations near the discontinuities while maintain stablesolution in the smooth areas. The unconditionally stable semi-implicit Crank-Nicolson (CN) scheme is used for time integration.The theoretical development was conducted based on one-dimensionalhomogeneous scalar equation and system equations. Discussions were alsoextended to include source terms and to deal with problems of higher dimension.For the treatment of source terms, Strang splitting was used. For multidimensionalequations, the ?? -form Douglas-Gunn alternating direction implicit(ADI) method was employed. To compare the performance of the scheme withENO type interpolation, the current numerical framework was also applied usingENO reconstruction.The numerical schemes were tested on 1-D and 2-D benchmark problems,as well as published experimental results. The simulated results show thecapability of the proposed scheme to resolve discontinuities while maintainingaccuracy in smooth regions. Comparisons with the experimental results validatethe method for dam break problems. It is concluded that the proposed scheme isa useful tool for solving hyperbolic equations in general, and from engineeringapplication perspective it provides a new way of modeling open channel flows.

This thesis examines the Euler equations of gas dynamics and develops a new adaption indicator, which is based on the weak local residual measured for the non- conservative pressure variable. We demonstrate that the proposed indicator is capable of automatically detecting discontinuities and distinguishing between the shock and contact waves when they are isolated from each other. We use the developed indi- cator to design a scheme adaption algorithm, according to which nonlinear limiters are used only in the vicinity of shocks. The new adaption algorithm is realized using a second-order limited scheme and a high-order nonlimited central-upwind scheme. Robustness and high resolution of the designed method is shown on a number of one- and two-dimensional numerical examples. / acase@tulane.edu

Numerical schemes for the partial differential equations used to characterize stiffly forced conservation laws are constructed and analyzed. Partial differential equations of this form are found in many physical applications including modeling gas dynamics, fluid flow, and combustion. Many difficulties arise when trying to approximate solutions to stiffly forced conservation laws numerically. Some of these numerical difficulties are investigated. A new class of numerical schemes is developed to overcome some of these problems. The numerical schemes are constructed using an infinite sequence of conservation laws. Restrictions are given on the schemes that guarantee they maintain a uniform bound and satisfy an entropy condition. For schemes meeting these criteria, a proof is given of convergence to the correct physical solution of the conservation law. Numerical examples are presented to illustrate the theoretical results.

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<p dir="ltr">This thesis introduces the evolving neural network method for solving scalar hyperbolic conservation laws. This method uses neural networks to compute solutions with an optimal moving mesh that evolves with the solution over time. The motivation for this method was to produce solutions with high accuracy near shocks while reducing the overall computational cost. The evolving neural network method first approximates initial data with a neural network producing a continuous piecewise linear approximation. Then, the neural network representation is evolved in time according to a combination of characteristics and a finite volume-type method.</p><p dir="ltr">It is shown numerically and theoretically that the evolving neural network method out performs traditional fixed-mesh methods with respect to computational cost. Numerical results for benchmark test problems including Burgers’ equation and the Buckley-Leverett equation demonstrate that this method can accurately capture shocks and rarefaction waves with a minimal number of mesh points.</p>

This thesis consists of an introduction and five papers concerning different numerical and mathematical aspects of viscous shocks. Hyperbolic conservation laws are used to model wave motion and advect- ive transport in a variety of physical applications. Solutions of hyperbolic conservation laws may become discontinuous, even in cases where initial and boundary data are smooth. Shock waves is one important type of discontinu- ity. It is also interesting to study the corresponding slightly viscous system, i.e., the system obtained when a small viscous term is added to the hyper- bolic system of equations. By a viscous shock we denote a thin transition layer which appears in the solution of the slightly viscous system instead of a shock in the corresponding purely hyperbolic problem. A slightly viscous system, a so called modified equation, is often used to model numerical solutions of hyperbolic conservation laws and their beha- vior in the vicinity of shocks. Computations presented elsewhere show that numerical solutions of hyperbolic conservation laws obtained by higher order accurate shock capturing methods in many cases are only first order accurate downstream of shocks. We use a modified equation to model numerical solu- tions obtained by a generic second order shock capturing scheme for a time dependent system in one space dimension. We present analysis that show how the first order error term is related to the viscous terms and show that it is possible to eliminate the first order downstream error by choosing a special viscosity term. This is verified in computations. We also extend the analysis to a stationary problem in two space dimensions. Though the technique of modified equation is widely used, rather little is known about when (for what methods etc.) it is applicable. The use of a modified equation as a model for a numerical solution is only relevant if the numerical solution behaves as a continuous function. We have experimentally investigated a range of high resolution shock capturing methods. Our experiments indicate that for many of the methods there is a continuous shock profile. For some of the methods, however, this not the case. In general the behavior in the shock region is very complicated. Systems of hyperbolic conservation laws with solutions containing shock waves, and corresponding slightly viscous equations, are examples where the available theoretical results on existence and uniqueness of solutions are very limited, though it is often straightforward to find approximate numerical solu- tions. We present a computer-assisted technique to prove existence of solu- tions of non-linear boundary value ODEs, which is based on using an approx- imate, numerical solution. The technique is applied to stationary solutions of the viscous Burgers' equation.We also study a corresponding method suggested by Yamamoto in SIAM J. Numer. Anal. 35(5)1998, and apply also this method to the viscous Burgers' equation.

<p>This thesis consists of an introduction and five papers concerning different numerical and mathematical aspects of viscous shocks. </p><p>Hyperbolic conservation laws are used to model wave motion and advect- ive transport in a variety of physical applications. Solutions of hyperbolic conservation laws may become discontinuous, even in cases where initial and boundary data are smooth. Shock waves is one important type of discontinu- ity. It is also interesting to study the corresponding slightly viscous system, i.e., the system obtained when a small viscous term is added to the hyper- bolic system of equations. By a viscous shock we denote a thin transition layer which appears in the solution of the slightly viscous system instead of a shock in the corresponding purely hyperbolic problem. </p><p>A slightly viscous system, a so called modified equation, is often used to model numerical solutions of hyperbolic conservation laws and their beha- vior in the vicinity of shocks. Computations presented elsewhere show that numerical solutions of hyperbolic conservation laws obtained by higher order accurate shock capturing methods in many cases are only first order accurate downstream of shocks. We use a modified equation to model numerical solu- tions obtained by a generic second order shock capturing scheme for a time dependent system in one space dimension. We present analysis that show how the first order error term is related to the viscous terms and show that it is possible to eliminate the first order downstream error by choosing a special viscosity term. This is verified in computations. We also extend the analysis to a stationary problem in two space dimensions. </p><p>Though the technique of modified equation is widely used, rather little is known about when (for what methods etc.) it is applicable. The use of a modified equation as a model for a numerical solution is only relevant if the numerical solution behaves as a continuous function. We have experimentally investigated a range of high resolution shock capturing methods. Our experiments indicate that for many of the methods there is a continuous shock profile. For some of the methods, however, this not the case. In general the behavior in the shock region is very complicated.</p><p>Systems of hyperbolic conservation laws with solutions containing shock waves, and corresponding slightly viscous equations, are examples where the available theoretical results on existence and uniqueness of solutions are very limited, though it is often straightforward to find approximate numerical solu- tions. We present a computer-assisted technique to prove existence of solu- tions of non-linear boundary value ODEs, which is based on using an approx- imate, numerical solution. The technique is applied to stationary solutions of the viscous Burgers' equation.We also study a corresponding method suggested by Yamamoto in SIAM J. Numer. Anal. 35(5)1998, and apply also this method to the viscous Burgers' equation.</p>

The class of hyperbolic conservation laws model the phenomena of non-linear wave propagation, including the presence and propagation of discontinuities and expansion waves. Such nonlinear systems can generate discontinuities in the so-lution even for smooth initial conditions. Presence of discontinuities results in break down of a solution in the classical sense and to show existence, weak for-mulation of a problem is required. Moreover, closed form solutions are di cult to obtain and in some cases such solutions are even unavailable. Thus, numerical algorithms play an important role in solving such systems. There are several dis-cretization techniques to solve hyperbolic systems numerically and Finite Volume Method (FVM) is one of such important frameworks. Numerical algorithms based on FVM are broadly classi ed into two categories, central discretization methods and upwind discretization methods. Various upwind and central discretization methods developed so far di er widely in terms of robustness, accuracy and ef-ciency and an ideal scheme with all these characteristics is yet to emerge. In this thesis, novel upwind and central schemes are formulated for various hyper-bolic systems, with the aim of maintaining right balance between accuracy and robustness. This thesis is divided into two parts. First part consists of the formulation of upwind methods to simulate genuine weakly hyperbolic (GWH) systems. Such systems do not possess full set of linearly independent (LI) eigenvectors and some of the examples include pressureless gas dynamics system, modi ed Burgers' sys-tem and further modi ed Burgers' system. The main challenge while formulating an upwind solver for GWH systems, using the concept of Flux Di erence Splitting (FDS), is to recover full set of LI eigenvectors, which is done through addition of generalized eigenvectors using the theory of Jordan Canonical Forms. Once the defective set of LI eigenvectors are completed, a novel (FDS-J) solver is for-mulated in such a manner that it is independent of generalized eigenvectors, as they are not unique. FDS-J solver is capable of capturing various shocks such as -shocks, 0-shocks and 00-shocks accurately. In this thesis, the FDS-J schemes are proposed for those GWH systems each of which have one particular repeated eigenvalue with arithmetic multiplicity (AM) greater than one. Moreover, each ux Jacobian matrix corresponding to such systems is similar to a unique Jordan matrix. After the successful treatment of genuine weakly hyperbolic systems, this strategy is further applied to those weakly hyperbolic subsystems which result on employ-ing various convection-pressure splittings to the Euler ux function. For example, Toro-Vazquez (TV) splitting and Zha-Bilgen (ZB) type splitting approaches to split the Euler ux function yield genuine weakly hyperbolic convective parts and strict hyperbolic pressure parts. Moreover, the ux Jacobian of each convective part is similar to a Jordan matrix with at least two lower order Jordan blocks. Based on the lines of FDS-J scheme, we develop two numerical schemes for Eu-ler equations using TV splitting and ZB type splitting. Both the new ZBS-FDS and TVS-FDS schemes are tested on various 1-D shock tube problems and out of two, contact capturing ZBS-FDS scheme is extended to 2-dimensional Euler system where it is tested successfully on various test cases including many shock instability problems. Second part of the thesis is associated with the development of simple, robust and accurate central solvers for systems of hyperbolic conservation laws. The idea of splitting schemes together with the notion of FDS is not easily extendable to systems such as shallow water equations. Thus, a novel central solver Convection Isolated Discontinuity Recognizing Algorithm (CIDRA) is formulated for shallow water equations. As the name suggests, the convective ux is isolated from the total ux in such a way that other ux, in present case other ux represents celerity part, must possess non-zero eigenvalue contribution. FVM framework is applied to each part separately and ux equivalence principle is used to x the coe cient of numerical di usion. CIDRA for SWE is computed on various 1-D and 2-D benchmark problems and extended to Euler systems e ortlessly. As a further improvement, a scalar di usion based algorithm CIDRA-1 is designed for v Euler systems. The scalar di usion coe cient depends on that particular part of the Rankine-Hugoniot (R-H) condition which involves total energy of the system as a direct contribution. This algorithm is applied to a variety of shock tube test cases including a class of low density ow problems and also to various 2-D test problems successfully. vi