Semi-analytic solutions to Riemann problem for one-dimensional gas dynamics

This work deals with the implementation of reduced differential transform method (RDTM) for solving the Riemann problem for gas dynamics in one dimension. The RDTM is an analytical method that can be applied to many linear and nonlinear partial differential equations and is capable of reducing the size of computations. Using this method, the solution is calculated in the form of convergent power series with easily computable components. The definition and basic properties of RDTM are investigated. Some new generalized formulas of reduced differential transforms are derived. The Riemann problem that describes the isentropic flow of an inviscid gas is considered to demonstrate the effectiveness and promising of the proposed algorithm.


INTRODUCTION
For one dimensional flows, the state U of a perfect gas is specified by two dependent variables, the velocity ( ) , u u x t = and the density ( ) of gas, and two convenient constants which are the ratio of specific heats γ and gas constant a .Hence the state U is completely defined by , , u v γ and a .
In this paper, the recently analytic technique, namely the reduced differential transform method (RDTM) (Keskin andOturanc, 2009, 2010), is presented for approximating the solutions to one-dimensional gas dynamics equations 2 1 0, 0, (1) That describes the isentropic flow of an inviscid gas (IFIG) (Gottlieb and Groth, 1988).Where c is the local speed.Given for perfect gas and the ratio of specific heats for air, we have: The system in Equation ( 1) is an example of a system of hyperbolic conservation laws for 0 a > and elliptic for 0 a < , but never to be of mixed type.

THE REDUCED DIFFERENTIAL TRANSFORM METHOD
To overcome the demerit of complex calculations of DTM, the RDTM was presented.This method was used by many mathematicians and engineers to solve various partial differential equations.The RDTM presents an efficient improvement in solving nonlinear partial differential equations since the amount of computations required is much less than that in other existing techniques.Its rapid convergence, gives exact solution with small number of iterations.
Consider the analytic and continuously differentiated function of two variables ( , ) u x t and suppose that it can be represented as a product of two single-variable functions, that is, Based on the properties of differential transform, the function ( , ) u x t can be represented as Where ( ) k U x is the transformed function, called t -dimensional spectrum function of ( , ) u x t , and defined by: Combining Equations ( 4) and ( 5) implies the differential inverse transform: One can easily obtained that this transform is derived from the power series expansion.Next, some basic theorems and generalized formulas of reduced differential transform are listed.
Theorem 1: The reduced differential transform is linear.

ANALYTIC SOLUTIONS OF ONE DIMENSIONAL PERFECT GAS EQUATIONS
The RDTM is designed to deal with continuous initial data.So, we will use the following transforms for our Riemann problem: , .In this part, the reduced differential transform technique is applied to solve IFIG model.Taking into account the local speed of gas flow in Equations ( 3) and ( 1) can be written in more compact form as: Operating the reduced differential transform for system in Equation ( 18), and using related facts in the previous section gives ( With starting transformed initial data ( ) ( ) ) 20), approximate solutions of order five are found with help of Mathematica.The obtained soliton solutions of velocity and density of gas are represented in Figures 1 and 2 respectively.In view, the methodology appears to be very promising for solving this system, it is convergent and stable.In this example, we cannot determine the errors in comparative to the exact solutions since we do not know these solutions.However, many terms can be calculated in order to achieve a high level of accuracy of the RDTM.

Conclusions
Many non-linear partial differential equations and systems that arising in various physical, chemical and engineering applications have no exact solution.So, semi-analytic solution for these equations is very important.In this paper, we found successfully approximate semi-analytic solutions for nonlinear one dimensional gas dynamics equations, namely the isentropic flow by an inviscid gas equations.The example presented demonstrates the fast convergence of the method.Another benefit of the reduced differential transform methodology is that it does not need any discretization to get numerical solutions.It introduces simple and straight forward calculations over other existing method.The numerical results obtained in this study show that the reduced differential transform method is powerful tool for solving linear nonlinear partial differential equations and systems.