The fourth order non oscillatory entropy stable scheme for degenerate convection diffusion equation part I

In this article, we formulate an entropy stable scheme for nonlinear degenerate convection diffusion equation with viscous terms in non-conservative formulations. Here in we extend the S.Jerez idea of first order entropy stable convection scheme for semi discrete scheme in to fourth order scheme using finite central difference scheme. Major advantage of this work is fourth order accuracy of the solution and fixed numerical diffusion term which can provide the nonoscillatory solution. Finally, few computational analyses are given to shown the accuracy of entropy stable scheme for degenerate parabolic equations.


Introduction:
In this section we discussed some basic definitions and results.

Notations of difference operator
We are discussing some general notations and basic formulas from [9]. The central difference with respect to t are defined by We need derive fourth order accurate aproximation to first and second derivative with respect tox ), ), ( ) 6 1 ( ), ( ), ( Rewrite above equation (7) is a positive semidefinite diffusion matrix defined in Ω. The diffusion term vanish in some sub spatial interval. For simplicity we denote = ( , ). This type of system represented by in porous media flow and two phase flow model [1][2][3][4][5].

If
= then equation (11) will be the following We replaced the value of L by first order forward difference operator for representing the first derivative. Substitute the equation (16) in to (14), then we get a fourth order scheme. Replace the notation = ∆ . The numerical scheme will be

Entropy stable scheme for degenerate convection diffusion equation 2.1. Basic results
Let us consider (12) with K → 0, then the equation have hyperbolic in nature. The main challenge for solving such type of equation is the shock wave formation near discontinuities of solution profile of nonlinear flux. Even for smooth initial data, the solution profile may have shock formation with nonlinear flux function. So we can use the weak formation in that situation. It is well known that weak solution do not converge to physically relevant unique solution. Lack of uniqueness of numerical solution must be enforced additional criteria, we call it as entropy condition. Next we discuss the entropy stability of the scheme (18). The accuracy and efficiency of entropy stable methods and higher order of accuracy discussed in [2,3,4] and discontinuous Galerkin methods in [5]. The entropy inequality for a degenerate parabolic problem Our main motivation of the study is to introduced the higher order entropy stable scheme. Entropy condition for system of equations (12) is defined in [6] Consider the three tuple (η,q,r) of functions from the set Ω to R, with strictly convex such that = , where the subscript w denotes the Hessian function. Recall that η = η(w) and q = q(w) are the entropy flux function, respectively, and the new function r = r(w) is named the diffusion entropy flux. Since η is strictly convex and v = η w is the entropy variable. Silvia Jerez Et.al. prove that degenerate form equation (12) is satisfy The equation (12) said to be entropy stable if there exist two numerical entropy fluxes Q j+1/2 and consistent with q and r x which satisfies the following condition (see [6])

Entropy stable scheme with Entropy conservative flux
Let us consider the Lefloch formula for higher order flux, then F 2p th-order accurate, in the sense that for sufficiently smooth solution w and entropy conservative by following discrete entropy equality.
is entropy stable provided the numerical viscosity matrix +1/2 satisfies the following conditions +1/2 +1/2 + proved in [6]. Here Jerez used entropy conservative flux, but due to the diffusion term numerical scheme become entropy stable provided k = 0.Suppose if k = 0 then scheme (29) will not be entropy stable. It satisfies only entropy conservative condition. In that case near discontinuity solution profile may exhibit Gibb's phenomenon near discontinuity region. To prevent this phenomenon, we need to add extra numerical diffusion term. Jerez modified the scheme for entropy stability that we discussed in next subsection.

Entropy stable scheme with Entropy stable flux
Consider the fourth order entropy stable flux defined in [2]. If K = 0 the numerical scheme equation will be , = − Here D is positive definite matrix. The scheme (32) using flux (33) is entropy stable by Cheng theorem 3.3 in [7] and results in [2]. Under the hypothesis of the theorem 3. ) , (34) Where >0 is proved by Jerez.

Fourth order entropy stable scheme for nondegenerate convection diffusion equation
The numerical flux +1/2 is defined by Existing tools for making higher order reconstruction is WENO-Z and JS-WENO e.t.c. But generally, it is not sign stable. Here we reconstruct.
[[v]] j+1/2 by fifth order WENO-Z method [8], but it is not sign stable. So we introduced limiter for preserving the sign stable.

Limiter for sign stability
In this article for reconstructing we used WENO-Z scheme, but it is not sign stable. So, we used by following way where s(j) is the numerical jump at x(j) ie., provided +1 + − − ≠ 0 otherwise s(i) = 0.
Here we consider the general Courant-Friedrich-Lewy(CFL) condition for parabolic equations is = max

4.Test problems
Consider the equation where w ∈ [0,∞] and diffusion matrix is defined as K(w) = µw 2 , and K 0 (w) = k(w). Based on the known entropy flux pair (η,g) for the burger equation. Entropy 3-tuple (η(w),g(w),r(w)) = is satisfied equation 1.11 in [6], where µ = 0.01. Let us consider entropy stable scheme for that an entropy conservative numerical flux is consider. w j 2 ) and a numerical viscosity matrix satisfying theorem 3.4 in [6] by For the computation we considering N = 1200for reference solution also. Mainly the following numerical methods are used for comparison ES1 flux is entropy conservative, nonconservative discretization of source term, numerical scheme is 29. ES2 flux is entropy stable, non-conservative discretization of source term using the numerical scheme is 34. ES3 Fourth order entropy stable scheme. All test problem using ES3 are SSP-RK4 method used for solving differential equations. Numerical simulations of ES1 and ES2 are obtained combining an entropystable spatial discretization with a TVD-RK2 time stepping. The errors and convergence rate of ES1 and ES2 are shown in [6][7][8][9]. It is first order accurate. If µ → 0 the PDE have hyperbolic in nature. So spurious oscillation in solution profile will produce the numerical scheme near discontinuity. From figure  1 we can understand after adding extra diffusion, oscillation removed. Next we are discussing the numerical result for non-oscillatory higher order entropy stable scheme.

Test problems for fourth order nonoscillatory entropy stable scheme
Recalling the equation (12). If µ tends to zero in (58) tends to hyperbolic case. Entropy conservative methods capture appearance of propagation of shock wave correctly but may produce strong oscillations in near shock region. In order to reduce the oscillation in [6] add some viscosity term. This work useful to capture correctly the non-classical shocks due to parabolic and hyperbolic interaction. For detailed reference see [6].
We used ES3 scheme with CFL=0.5, and Final time T = 1; Rate of convergence and accuracy table of the problem 57 is given.

Conclusion
Herein we introduce the concept of fourth order entropy stable scheme for degenerate convection equation and new limiter for preserving sign stability for numerical diffusion term. Using this limiter we can use all type of higher order WENO for reconstruction of diffusion term, Because of this we can generate a higher order nonoscillatory entropy stable scheme.