Numerical Method for Non-Linear Conservation Laws: Inviscid Burgers Equation

This paper deals with the Burgers equation which is the most common model used in the nonlinear conservation laws. Here the theoretical aspect of conservation law is discussed by using inviscid Burgers equation. At first, we in-troduce the general non-linear conservation law as a partial differential equation and its solution procedure by the method of characteristic. Next, we present the weak solution of the problem with entropy condition. Taking into account shock wave and rarefaction wave, the Riemann problem has also been discussed. Finally, the finite volume method is considered to approximate the numerical solution of the inviscid Burgers equation with continuous and discontinuous initial data. An illustration of the problem is pro-vided by some examples. Moreover, the Godunov method provides a good approximation for the problem.


Introduction
The Nonlinear conservation law is used to describe many physical phenomena mathematically, and numerical methods are required to solve these physical problems [1] [2] [3] [4]. The governing one-dimensional conservation law takes the form [5], where f(u) is the flux function. The quantity u is the density of some fluid and by the volume of the cell. On the other hand, in finite difference method, the pointwise approximation is needed at the grid point. The cell average is changed in each time step by the flux through the edges of the grid cell. Therefore, it is important to choose a good numerical flux function that approximates the correct fluxes reasonably well [5]. Therefore finite volume method is effective to describe the discontinuous solution. The main objective of this paper is to approximate the numerical solution of the non-linear conservation law by using the inviscid Burgers equation. In this paper, Godunov method is considered to approximate the numerical solution of the problem with continuous and discontinuous initial data. The Godunov method is a conservative numerical method, which is used for the solution of hyperbolic conservation laws. This method can be characterized by the solution (exact or approximate) of a Riemann problem within computational cells in order to obtain numerical fluxes. This method is first-order accurate in both space and time.

Integral Form of Conservation Laws
The integral form of the conservation law is obtained by integrating (1) within , x x and defined as, ( ) ( ) The above equation is the integral form of the conservation law which directly interpreted that the difference in the total amount of the state variable u in [ ]

Classical Solution
To understand the classical solution, initially we will consider simple linear conservation law, known as linear advection equation [6] where ( ) The solution of advection equation can be easily derived from the concept of method of characteristic. Here, the characteristic curve = , describe the straight line with slope a in x = x(t) curve and the solution is Thus, linear conservation law transport the initial data in the x-direction and also it is exactly as smooth as the initial data.
In this research, Burgers equation with zero viscosity term is considered which is known as inviscid Burgers equation. The inviscid Burgers equation is a basic case study as it has also the properties of nonlinear conservation law [11]. Therefore inviscid Burgers equation takes the form, The characteristic curve of the equation is The characteristics are straight lines with slope u, and the solution is Thus, the solution is constant along the characteristic curves. In this case, discontinuity may form smooth initial conditions in finite time. Therefore, the study of solution of Burgers equation allows interesting phenomena such as shock waves and rarefaction waves.

Weak Solution
In classical sense, if u be a piecewise constant solution of the conservation law Journal of Applied Mathematics and Physics (1) and also discontinuous along the characteristic curve with any shock speed (s) then u is a weak solution of (1). The shock speed can be derived from the Rankine-Hugoniot jump condition [5] [12]. To get the shock speed, we need to Suppose shock splits the rectangle into two parts and L u and R u are two states immediately to the left and right side of the shock, then Which is known as the Rankine-Hugoniot Condition and s is the shock speed.
The shock speed for Inviscid Burgers equation is Therefore it is easy to calculate the shock speed. In general, weak solutions occur whenever there is no smooth or classical solution and these solutions may not be differential or even continuous. Therefore, in this case, Equation (1) is not valid and the integral form of conservation law does hold. In mathematically, the function ( ) , u x t is a weak solution of conservative law (1) with initial condition if it satisfies the following, for all function of ( ) However, the weak solution concept cannot guarantee the uniqueness of the solution, because of that addition condition need to be imposed, called entropy condition.

Entropy Condition
The entropy condition described by the following [12], If a discontinuity propagate with the characteristic speed s, given by the Rankine-Hugoniot condition, satisfies the entropy condition if, In particular, for the inviscid Burgers equation, if u satisfy the entropy condition then,

Riemann Problem
The Riemann problem is a particular initial value problem which contains a conservation law together with piecewise constant data having a single discontinuity. We consider the Riemann problems for the inviscid Burgers equation subject to special initial conditions, described by left wave travel faster than the right wave and as a result, a shock wave is generate. In this case, the solution satisfies the entropy condition also.
On the other hand, if L R u u < , the problem has two solutions. First one is, But this solution does not follow the entropy condition, which is described above [12].
Another solution is weak solution. To get the weak solution suppose , then the problem becomes, u < , the characteristic waves are spreading and produced rarefaction wave, head of the wave is faster than tail. Figure 2 shows the rarefaction wave.
The study of Riemann problem is very useful for understanding shock wave and rarefaction wave which appear as characteristics in the solution. In numerical methods, Riemann problem is also the basic tool for developing finite volume methods for the solution of conservation law equations due to the discreteness of the grid.
The Riemann problem has a similarity solution, a function of x/t alone; selfsimilar at different times. The solution ( ) ( ) , u x t u x t = is constant along any ray x/t = constant through the origin.

Numerical Method
In this paper, finite volume method is considered for the numerical solution of the one dimensional conservation law. Initially, we have discussed the discretization the computational domain in both space and time of the problem (1).
Thus, the i-th grid cell or control volume is ( ) The discretization in time with step size ∆t is n t n t = ∆ . Now, suppose the value n i u is the cell average over the i-th interval at time t n .
Then,  In finite volume method, different way of approximation of flux function will give the different methods. Next, the Godunov method is presented by approximating flux function.

Godunov Method
For Godunov method, we need to approximate the numerical flux Where s is the shock speed and The formula can also be written as [5] [14], Additionally, it guarantees that the solution satisfies the entropy condition. However, the necessary condition for consistency of the numerical flux function is that the wave speed should be bounded [14], i.e.

( )
Moreover, the CFL condition for the given problem is

Examples by Godunov Method
Here, the initial value problem for inviscid Burgers equation using Godunov

Discontinuous Initial Value Problems
Initially considering inviscid Burgers equation with the following discontinuous initial value problem, The initial condition and the numerical solution using Godunov method is presented in Figure 3 and Again, considering the following discontinuous initial value problem,  The initial condition and the numerical solution shows the rarefaction wave using Godunov method is presented in Figure 5 and Figure 6, where, 0.005 t ∆ = and 0.01 x ∆ = . Figure 6 also shows the wave at different time, where n is represent the number of iteration. The computational time is 0.40 second.

Continuous Initial Value Problems
Next, considering inviscid Burgers equation with following continuous initial condition is as smooth as possible. Figures 7-10 show the initial conditions and the numerical solutions by Godunov method of (4) and (5) respectively, where, x -axis is only differ from other.
In inviscid Burgers equation is u itself. In this case, the initial value moves with variable speed in x. As the sine wave is positive at 0 x ≤ ≤ π , and negative at 2 x π ≤ ≤ π , the solution is therefore moves first forward and the backward direction, and develop a discontinuity at x = π .      In Godunov method, the solution computed on a very fine mesh. Figure 8 shows initially sine wave compress and develop a shock, and then it expands. In Figure 10, the solution is called N-wave also. Therefore, Godunov method shows a good approximation of the solution.

Conclusion
In this research, inviscid Burgers equation is considered as an example of nonlinear conservation law. Here we presented a theoretical feature of nonlinear conservation law by using the common model, inviscid Burgers equation. Some important features such as integral form of conservation law, classical solution, and weak solution with entropy condition also have been presented. The Riemann problem is discussed to understand the shock wave and rarefaction wave.
Finite volume method is considered as numerical method to approximate the solution, as it is effective to describe the discontinuous solution. Taking into account a special initial value problem such as continuous and discontinuous initial value problems are solved by Godunov method. In conclusion, Godunov method shows a good approximation of the problem.