Adaptive Mesh Refinement (AMR) is a powerful tool for computing solutions to partial Differential Equations (PDEs) whose solutions have disparate spatial scales [1]. Currently, several different AMR strategies have emerged. These approaches can be classified into four broad categories depending on the partitioning algorithm used and/or the data structure that is adopted to keep track of the mesh connectivity. They are as follows: (1) ―patch-based‖, (2) ―cell-based‖, (3) ―block-based‖, (4) ―hybrid block-based‖ AMR techniques. In this paper an algorithm for dynamic gridding is presented. It was first proposed by Berger, Oliger and Colella [2, 3], now more generally referred to as patch-based AMR. The algorithm begins with the entire computational domain covered with a coarsely resolved base-level regular Cartesian grid. As the calculation progresses, individual grid cells are tagged for refinement. The patch-based AMR strategy relies on a fairly sophisticated algorithm, laid out by Berger, to organize a collection of individual grid cells into properly nested rectangular patches. The mesh within these newly farmed patches can then be further refined, creating additional patches. The fundamental computational problem in AMR consists of constructing, managing and carrying out the solution process on a hierarchy of grids [4, 5, 6]. This requires that we are able to express the AMR application in terms of familiar abstractions that are natural to the process of solving a self-adaptive grid hierarchy. Associated with this grid is some data that is specific to the particular solution process that is being used. The solution process applies one or more numerical algorithms to this data. The algorithm of adaptive refinement is applied to solve the following equations in one-dimensional setting: transfer equation 0. u u c t x (1) Burgers’ equation 0. u u u t x (2) And the equations of hydrodynamics 0, 0, 0. i i i i j ij i i i v t x v v v P t x e e P v t x (3) Transfer equation and Burgers’ equation are solved with the help of upwind, Lax and Lax-Wendroff schemes. Nonlinear second-order accurate Total Variation Diminishing (TVD) scheme is used to solve Euler equations. It is build upon the first-order monotone upwind scheme. The second-order accurate fluxes 1/2 t n F at cell boundaries are obtained by taking first-order fluxes (1), 1/2 t n F from the upwind scheme and modifying it with a second order correction. The first-order upwind flux (1), 1/2 t n F comes from the averaged flux t n F in cell n . The second-order flux corrections are defined using three local cell-centered fluxes and a flux limiter . Three TVD limiters are used: 1. The minmod flux limiter chooses the smallest absolute value between the left and right corrections. 2. The superbee limiter chooses between the larger correction and two times the smaller correction, whichever is smaller in magnitude. 3. The Van Leer limiter takes the harmonic mean of the left and right corrections.