THIRD ORDER NON-OSCILLATORY CENTRAL SCHEME FOR MULTIDIMENSIONAL HYPERBOLIC CONSERVATION LAWS

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1 Palestine Journal of Mathematics Vol. 5(Special Issue: ) (6), 8 37 Palestine Polytechnic University-PPU 6 THIRD ORDER NON-OSCILLATORY CENTRAL SCHEME FOR MULTIDIMENSIONAL HYPERBOLIC CONSERVATION LAWS A. A. I. Peer, A. A. E. F. Saib, M. S. Sunhaloo and M. Bhuruth Communicated by Suheil Khoury MSC Classifications: Primary 65M8; Secondary 35L65. Keywords and phrases: Multidimensional hyperbolic conservation laws, non-oscillatory central scheme, non-linear limiters. Abstract. We derive a third-order non-oscillatory central scheme for multidimensional hyperbolic systems. We use computationally efficient limiters in order to make the new scheme non-oscillatory. The solution is advanced in time using natural continuous extension of Runge- Kutta methods. We pay particular attention to two dimensional problems including inviscid Burgers equation and Riemann gas dynamics problems. Numerical experiments show that the new scheme remains non-oscillatory while giving good resolution of discontinuities. Introduction Multidimensional hyperbolic conservation laws of the form u t + x f(u)=, x R d, (.) and with initial datau(x, t = ) = u (x), have been used as models for a wide variety of physical phenomena. Examples include the Buckley-Leverett equation for modelling a one-dimensional two-phase flow of immiscible fluids through porous media [3] and the Euler equations of gasdynamics [5, 4]. Other examples can be found in the book by [7]. Some major developments in central schemes for multidimensional problems include nonoscillatory piecewise polynomial reconstructions by [5, 6]. A class of multidimensional central weighted essentially non-oscillatory (CWENO) methods was developed in [6, 5, 7]. These schemes enjoy a black-box approach, that is, only the flux is required and can be extended to most hyperbolic problems, e.g. [8]. A family of semi-discrete central schemes for solving multidimensional problems has been proposed in [8,,, 9], which are advanced in time with ODE solvers. In this work, we develop a third order non-oscillatory central scheme for approximating multidimensional hyperbolic conservation laws (.). The new scheme is genuinely non-oscillatory in the sense of [, ] compared to ENO reconstructions. It combines the minmod limiter with a quadratic polynomial. Though, the scheme can be applied as a black-box solver, we will be paying special attention to the numerical solution of multidimensional inviscid Burgers equation and Euler equations of gas dynamics. The outline of this paper is as follows: In Sect., the general approach for solving multidimensional problems using staggered central schemes is presented. The reconstruction of our new scheme is discussed in Sect. 3. In Sect. 4 some numerical examples are given and solved with the proposed method. The paper ends with conclusion in Sect. 5. Multi-Dimensional Central Schemes For simplicity, we consider the two-dimensional problem, where d = in (.) u t +f(u) x +g(u) y =. (.)

2 THIRD ORDER NON-OSCILLATORY CENTRAL SCHEME 9 We further consider a uniformly discretised cell-centered computational domain where I ij = [x i, x i+ ] [y j, y j+ ] and the width in the x and y directions are respectively denoted by h and k. For simplicity, we describe the case when h = k. The cell-centres are given by(x i, y j ) where x i = (x i +x i+ ) and y j = (y j +y j+ ). In central schemes, the cell-averages at time t n given by ū n ij = h I ij u(x, y, t n )dydx, are assumed to be known. Then a piecewise polynomial interpolating the cell averages is used u n (x, y) = i,j p n i,j(x, y)χ i,j (x, y), (.) where χ ij (x, y) is the characteristic function of the cell I ij. Integrating (.) over I i+,j+ [tn, t n+ ], gives the fully discrete scheme ū n+ i+,j+ =ū n i+,j+ t n+ h τ=t n t n+ h τ=t n yj+ y=y j [f (u(x i+, y, τ)) f (u(x i, y, τ))]dydτ xi+ x=x i [g(u(x, y j+, τ)) g(u(x, y j, τ))]dxdτ where the staggered cell average ū n are obtained from (.) as follows: i+,j+ ū n i+,j+ = h = h + I i+,j+ xi+ yj+ x i xi+ yj+ x i+ y j y j u n (x, y)dydx xi+ p n yj+ i,j(x, y)dydx+ p n i,j+(x, y)dydx x i y j+ xi+ yj+ p n i+,j(x, y)dydx+ p n i+,j+(x, y)dydx x i+ y j+ } }, (.3). (.4) The integrals of (.3) are evaluated by reconstructing the point values of u(x, y, τ) t n τ t n+ } from the known cell-averages. Then, for a sufficiently small time step t, the integrals are assumed to be smooth, such that appropriate quadrature rules can be used. For example, one can use the following quadrature rule in space: xi+ x i f(x)dx = h 4 [ f(x i+)+3f(x i+ )+3f(x i ) f(x i )]+O(h 4 ), (.5) and the Simpson s rule given by: t n+ τ=t n f (u(x i, y j, τ))dτ = t m γ l [f (u(x i, y j, t n +β l t))], (.6) l= to evaluate the time integrals where β =, β = /, β =, γ = /6, γ = /3 and γ = /6. For this quadrature rule, we need to predict the intermediate values u(x i, y j, t n+/ ) and u(x i, y j, t n+ ). Once more, we use the smoothness of the approximations along the lines (x i, y j ) [t n, t n+ ] to consider the Cauchy problem: v i,j(τ) = F(τ, v i,j (τ)) = f x (v(x i, y j, t n +τ)) g y (v(x i, y j, t n +τ)), v i,j (t ) = u(x i, y j, t n ). (.7)

3 3 Peer, Saib, Sunhaloo and Bhuruth Similar to [, ], we use the Natural Continuous Extension (NCE) of Runge-Kutta (RK) methods [6] to solve the problem up to τ = t. We use the second-order RK method given by the polynomials with the following set of coefficients: ( ) / b =, a = / b (θ) = (b )θ +θ, (.8) b (θ) = b θ, (.9) ( ). (.) To compute the predicted values of the quadrature formula u(x i,y j,t n + β l t) efficiently, we rewrite the NCE-RK method as u(x i, y j, t n +β l t) = u n ij +λ b r (β l )Kij r, r= K r ij = f x (Y r ij) g y (Y r ij), (.) r ij = u n ij +λ a rs Kij s, Y r where the coefficientsa ij are given in (.), and the numerical derivativef x andg y are discussed in the next section. 3 A Third-Order Non-Oscillatory Central Scheme 3. Reconstruction from Cell Averages We first reconstruct the piecewise-polynomials p n i,j (x, y) of (.). Following [6], we consider a polynomial of degree associated with the cell I i,j : p n i,j(x, y) =u n i,j +u xi,j ( x xi h ) +u yi,j ( y yj h + h ( u xxi,j (x x i ) + u xyi,j (x x i )(y y j )+u yyi,j (y y j ) ). (3.) We require that p n i,j (x, y) obeys the conservation property: h p n i,j(x, y)dydx = ū n i,j, I ij that is,u n i,j must satisfy u n i,j = ū n i,j ) (u xxi,j +u yyi,j. (3.) 4 We choose the numerical derivatives to be fully non-oscillatory as [5], compared to the CWENO reconstructions of [6] and [7]. In addition, they must yield third-order accuracy, that is, s= ) h u x i,j = x u(x = x i, y = y j, t n )+O(h ), (3.3) h u y i,j = y u(x = x i, y = y j, t n )+O(h ), (3.4) h u xx i,j = x u(x = x i, y = y j, t n )+O(h), (3.5) h u yy i,j = y u(x = x i, y = y j, t n )+O(h), (3.6) h u xy i,j = x y u(x = x i, y = y j, t n )+O(h). (3.7)

4 THIRD ORDER NON-OSCILLATORY CENTRAL SCHEME 3 Using (3.) and (3.), the reconstruction of (.4) gives ū n i+,j+ = 4 (ūn i,j +ū n i+,j +ū n i,j+ +ū n i+,j+) + 6 (u x i,j u xi+,j +u xi,j u xi+,j+ ) + 6 (u y i,j +u yi+,j u yi,j u yi+,j+ ) + 64 (u xy i,j u xyi+,j u xyi,j +u xyi+,j+ ). (3.8) In order to achieve third-order accuracy for (3.8), we use the UNO limiter to find the numerical derivatives (3.3), (3.4) and (3.7): u xi,j = MM ū i,j + MM ū i,j, } ū i,j, ūi+,j MM ū i,j, } } ū i+,j, (3.9) u yi,j = MM ū i,j + MM ū i,j, } ū i,j, ūi,j+ MM ū i,j, } } ū i,j+, (3.) u xyi,j = MM u + yi,j MM u yi,j, } u yi,j, uyi+,j MM u yi,j, } } u yi+,j. (3.) One may alternatively find u xyi,j of (3.), by using the derivatives on the y axis of u xi,j. Numerical experiments indicate that similar results are obtained. The NT scheme [] uses a second-order accurate limiter ) v j = MM ( v j, v j+, (3.) which is non-oscillatory in the sense that ( ) v j. sign( v j± ) Const. MM v j, v j+ Here, v j+ = v j+ v j, and the MinMod limiter (MM) is defined by min p x p } if x p > p, MM(x, x,...) = max p x p } if x p < p, otherwise. However, the accuracy of (3.) drops at the non-sonic critical gridvalues v j, where v j v j+ < f (v j ). NT scheme adapted the uniform non-oscillatory (UNO) limiter of Harten and Osher [4] ( v j = MM v j + MM( v j, ) v j, vj+ MM( v j, ) ) v j+, (3.3) where v j = v j+ v j +v j. The limiter (3.3) adds second-order differences to the MinMod limiter (3.) to achieve high accuracy including at critical points. 3. Reconstruction from Point Values We approximate the point-value u n i,j of (3.) from the cell averages ū n i,j, while simultaneously looking for high accuracy and avoiding oscillations. This is achieved by approximating the derivatives of (3.) by u xxi,j = MM ( ū n i,j, ū n i,j, ū n i+,j), (3.4) u yyi,j = MM ( ū n i,j, ū n i,j, ū n i,j+). (3.5)

5 3 Peer, Saib, Sunhaloo and Bhuruth Following [], we compute the MM limiter with three arguments as follows MM(x, x, x 3 ) = 4 (sign(x )+sign(x )+sign(x 3 )+sign(x x x 3 )) min( x, x, x 3 ). In order to compute the fluxes of (.7), we must evaluate the function F(u i,j ) = f x (u i,j ) g y (u i,j ), (3.6) where u i,j is found at the intermediate time intervals given by the NCE-RK method, notably for u(x i, y j, t n+/ ) and u(x i, y j, t n+ ). The numerical derivatives of (3.6) must satisfy thirdorder accuracy, that is, h f x(u i,j ) = x f(u(x= x i, y = y j, t n ))+O(h ), (3.7) h g y(u i,j ) = y g(u(x = x i, y = y j, t n ))+O(h ). (3.8) We approximate f x (u i,j ) and g y (u i,j ) in a similar way as in (3.9) and (3.) by using the UNO limiter on differences of f and g to obtain the desired order of accuracy and still remain nonoscillatory. The derivatives of the fluxes are then given by f xi,j = MM f i,j + MM f i,j, } f i,j, fi+,j MM f i,j, } } f i+,j, (3.9) g yi,j = MM g i,j + MM g i,j, } g i,j, gi,j+ MM g i,j, } } g i,j+. (3.) 4 Numerical Experiments Problem We test the accuracy of the third-order scheme, which we call CNO3, on the D linear advection problemu t +u x +u y = with the initial conditionu(x, y, ) = sin (πx) sin (πy) and periodic boundary conditions on the domain [, ] [, ]. In Table we give the error and numerical order obtained on different grid sizes with λ =.3 at T =. We see that the scheme is converging to third-order accuracy, but it does not achieve the maximum order of accuracy due to the non-smooth limiters [4]. For comparison, we include the results obtained by the thirdorder central scheme with CWENO weights and constant weights of [6] for λ =.45. We see that those schemes give comparable convergence to our proposed scheme, but smaller magnitude of L errors. Similar observations are made with the compact scheme of [5]. However, the fourth-order central scheme of [7] performs better as expected. Problem We consider the linear rotation problem of [7] whereu t +f(u,x,y) x +g(u,x,y) y = on the square patch [, ] [, ] with N = 4 in each direction and λ =.3. The initial condition and fluxes are respectively given by, x u (x, y) = < and y <,, otherwise, and ( f(u,x,y)= y ) ( πu, g(u,x,y)= x ) πu. Figure shows the solutions at T =.5 and T = by the CNO3 scheme which are free from spurious oscillations.

6 THIRD ORDER NON-OSCILLATORY CENTRAL SCHEME 33 Table. Accuracy by central schemes with u(x, y, ) = sin (πx) sin (πy). N CNO3 L error L error L error L error.4988(-) (-).8.535(-) (-3) (-) (-5) (-3).885 N Constant weights C-WENO weights L error L order L error L order.57(-).854(-).667(-) (-) (-3) (-) (-4) (-4) (a) T =.5 (b) T = Figure. Solution of Problem by CNO3, with N = 4 and λ =.3.

7 34 Peer, Saib, Sunhaloo and Bhuruth u(x,y,).4. u(x,y,) y.5.75 x y.5.75 x.5 (a) T = (b) T = u(x,y,3).4. u(x,y,4) y.5.75 x y.5.75 x.5 (c) T = 3 (d) T = 4 Figure. Solution of Burgers equation by CNO3 with λ =.3 and h =.. Problem 3 We next solve the Burgers equation u t + ( u ) x + ( u ) y = for the initial condition from [6],u(x, y, ) = sin (πx) sin (πy), on the domain [, ] [, ] with periodic boundary conditions. In Figure, we display the results up to T = 4, on 5 5 grid with λ =.3. We observe that the solutions obtained by CNO3 are well resolved and non-oscillatory. Problem 4 Finally, we test CNO3 on the Riemann problems for two-dimensional gas dynamics U t +F(U) x +G(U) y =, where U = ρ ρu ρv E, F(U) = ρu ρu +p ρuv u(e +p), G(U) = ρu ρuv ρv +p v(e +p),

8 THIRD ORDER NON-OSCILLATORY CENTRAL SCHEME 35 and E = p (γ ) + ρ(u +v ). Here ρ is the density, u and v are the velocity components, E is the total energy and p is the pressure. γ =.4 is the ratio of specific heats. Nineteen configurations were identified for those problems [, 3, 3, ], where the initial condition on [, ] [, ] is given by (p, ρ, u, v)(x, y, ) = (p, ρ, u, v ), if x >.5, y >.5, (p, ρ, u, v ), if x <.5, y >.5, (p 3, ρ 3, u 3, v 3 ), if x <.5, y <.5, (p 4, ρ 4, u 4, v 4 ), if x >.5, y <.5. The different quadrants are initially separated by either rarefaction, shock or contact wave. We tested all the different configurations using componentwise extension with λ =.3 and grid but due to space constraints we show only a few results in Figure 3. We observe that CNO3 recovers the major features on the density profiles in most test cases, but is somewhat smeared near discontinuities (a) Configuration : T = (b) Configuration : T = (c) Configuration 4: T = (d) Configuration 9: T =.3. Figure 3. Approximation of some two-dimensional gas dynamics problems by CNO3

9 36 Peer, Saib, Sunhaloo and Bhuruth 5 Conclusion In this work, we have introduced a genuinely multidimensional third-order non-oscillatory central scheme. A piecewise quadratic polynomial is used for the reconstruction, which uses highorder accurate approximation for the spatial derivatives to avoid spurious oscillations. The scheme was tested on different problems and we observed that it captured the right profile of the solutions. However, the scheme is damped and discontinuities were smeared due to the nonlinear limiter. Alternatively, a sharper non-oscillatory multi-dimensional reconstruction may be used, but this will necessitate additional numerical derivatives. The proposed scheme can also be extended to solve Hamilton-Jacobi equations [9] and MHD equations []. References [] J. Balbas, E. Tadmor, and C.-C. Wu. Non-oscillatory central schemes for one-and two-dimensional MHD equations: I. J. Comput. Phys., ():6 85, 4. [] F. Bianco, G. Puppo, and G. Russo. High-order central schemes for hyperbolic systems of conservation laws. SIAM J. Sci. Comput., ():94 3, 999. [3] M. Geiben, D. Kröner, and M. Rokyta. A Lax-Wendroff type theorem for cell-centered, finite volume schemes in -D. preprint, 78(SFB56), 99. [4] A. Harten and S. Osher. Uniformly high-order accurate nonoscillatory schemes, I. SIAM J. Numer. Anal., 4():79 39, 987. [5] G. S. Jiang and E. Tadmor. Nonoscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput., 9(6):89 97, 998. [6] T. Katsaounis and D. Levy. A modified structured central scheme for D hyperbolic conservation laws. Applied mathematics letters, (6):89 96, 999. [7] D. Kröner. Numerical Schemes for Conservation Laws. Wiley Teubner, Chichester, 997. [8] A. Kurganov and D. Levy. A third-order semidiscrete central scheme for conservation laws and convection-diffusion equations. SIAM J. Sci. Comput., (4):46 488,. [9] A. Kurganov, S. Noelle, and G. Petrova. Semidiscrete central-upwind schemes for hyperbolic conservation laws and hamilton-jacobi equations. SIAM J. Sci. Comput., 3(3):77 74,. [] A. Kurganov and G. Petrova. A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems. Numer. Math., 88:683 79,. [] A. Kurganov and E. Tadmor. New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys., 6:4 8,. [] A. Kurganov and E. Tadmor. Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Numer. Meth. Partial Differ. Eq., 8:584 68,. [3] P. D. Lax and X.-D. Liu. Solution of two-dimensional Riemann problems of gas dynamics by positive schemes. SIAM J. Sci. Comput., 9:39 34, 998. [4] R. J. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge, UK,. [5] D. Levy, G. Puppo, and G. Russo. Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput., ():656 67,. [6] D. Levy, G. Puppo, and G. Russo. A third order central WENO scheme for D conservation laws. Appl. Numer. Math., 33:45 4,. [7] D. Levy, G. Puppo, and G. Russo. A fourth-order central WENO scheme for multidimensional hyperbolic systems of conservation laws. SIAM J. Sci. Comput., 4():48 456,. [8] G. Li, V. Caleffi, and J. Gao. High-order well-balanced central WENO scheme for pre-balanced shallow water equations. Computers and Fluids, 99:8 89, 4. [9] C.-T. Lin and E. Tadmor. High-resolution nonoscillatory central schemes for hamilton jacobi equations. SIAM J. Sci. Comput., (6):63 86,. [] H. Nessyahu and E. Tadmor. Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys., 87():48 463, 99. [] A. A. I. Peer, A. Gopaul, M. Z. Dauhoo, and M. Bhuruth. A new fourth-order non-oscillatory central scheme for hyperbolic conservation laws. Appl. Numer. Math., 58: , 8. [] C. W. Schulz-Rinne. Classification of the Riemann problem for two-dimensional gas dynamics. SIAM J. Math. Anal., 4:77 88, 993.

10 THIRD ORDER NON-OSCILLATORY CENTRAL SCHEME 37 [3] C. W. Schulz-Rinne, J. P. Collins, and H. M. Glaz. Numerical solution of the Riemann problem for two-dimensional gas dynamics. SIAM J. Sci. Comput., 4:394 44, 993. [4] S. Serna and A. Marquina. Power ENO methods: a fifth-order accurate Weighted Power ENO method. J. Comput. Phys., 94:63 658, 4. [5] E. F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer-Verlag, Berlin, 999. [6] M. Zennaro. Natural continuous extensions of Runge-Kutta methods. Math. Comput., 46:9 33, 986. Author information A. A. I. Peer, A. A. E. F. Saib, M. S. Sunhaloo, Department of Applied Mathematical Sciences, University of Technology, Mauritius, La Tour Koenig, Pointe-aux-Sables 34, Mauritius. Ô ÖÙÑ ÐºÙØѺ ºÑÙ ÙÑ ÐºÙØѺ ºÑÙ ÙÒ ÐÓÓÙÑ ÐºÙØѺ ºÑÙ M. Bhuruth, Department of Mathematics, University of Mauritius, Reduit 8837, Mauritius. Ñ ÙÖÙØ ÙÓѺ ºÑÙ Ê Ú ÔØ Å Ý ½¼ ¾¼½ º ÂÙÐÝ ½½ ¾¼½