In this article, the two-phase water hammer theoretical and numerical simulation are provided. A mathematical formulation is presented to describe transient one-dimensional flow of bubbly gas-liquid mixtures without phase change in an horizontal pipe. The features two-fluid model for simulating flows investigated. governing equations were obtained from mass momentum conservation laws combined w...

We present a new family of high-order shock-capturing finite difference numerical methods for systems conservation laws. These methods, called Adaptive Compact Approximation Taylor (ACAT) schemes, use centered $(2p + 1)$-point stencils, where $p$ may take values in $\{1, 2, \dots, P\}$ according to smoothness indicators the stencils. The are based on combination robust first order scheme and Ap...

Single-stage or single-step high-order temporal discretizations of partial differential equations (PDEs) have shown great promise in delivering accuracy time with efficient use computational resources. There has been much success developing such methods for finite volume method (FVM) PDEs. The Picard Integral formulation (PIF) recently made single-stage accessible difference (FDM) discretizatio...

In [18], two of the authors developed a high order accurate numerical boundary condition procedure for hyperbolic conservation laws on a Cartesian mesh, which allows the computation using high order finite difference schemes on Cartesian meshes to solve problems in arbitrary physical domains whose boundaries do not coincide with grid lines. This procedure is based on the so-called inverse Lax-W...

We develop a high order finite difference numerical boundary condition for solving hyperbolic conservation laws on a Cartesian mesh. The challenge results from the wide stencil of the interior high order scheme and the fact that the boundary intersects the grids in an arbitrary fashion. Our method is based on an inverse Lax-Wendroff procedure for the inflow boundary conditions. We repeatedly us...

Goal of this thesis is to study four problems. In chapters 3-5, we consider scalar conservation law in one space dimension with strictly convex flux. First problem is to know the profile of the entropy solution. In spite of the fact that, this was studied extensively in last several decades, the complete profile of the entropy solution is not well understood. Second problem is the exact control...