Novel Upwind and Central Schemes for Various Hyperbolic Systems

Abstract

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

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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.

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