THIRD ORDER NON-OSCILLATORY CENTRAL SCHEME FOR MULTIDIMENSIONAL HYPERBOLIC CONSERVATION LAWS
|
|
- Briana Gilmore
- 5 years ago
- Views:
Transcription
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
A New Fourth-Order Non-Oscillatory Central Scheme For Hyperbolic Conservation Laws
A New Fourth-Order Non-Oscillatory Central Scheme For Hyperbolic Conservation Laws A. A. I. Peer a,, A. Gopaul a, M. Z. Dauhoo a, M. Bhuruth a, a Department of Mathematics, University of Mauritius, Reduit,
More informationA Fourth-Order Central Runge-Kutta Scheme for Hyperbolic Conservation Laws
A Fourth-Order Central Runge-Kutta Scheme for Hyperbolic Conservation Laws Mehdi Dehghan, Rooholah Jazlanian Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University
More informationSolution of Two-Dimensional Riemann Problems for Gas Dynamics without Riemann Problem Solvers
Solution of Two-Dimensional Riemann Problems for Gas Dynamics without Riemann Problem Solvers Alexander Kurganov, 1, * Eitan Tadmor 2 1 Department of Mathematics, University of Michigan, Ann Arbor, Michigan
More informationA FOURTH-ORDER CENTRAL WENO SCHEME FOR MULTIDIMENSIONAL HYPERBOLIC SYSTEMS OF CONSERVATION LAWS
SIAM J. SCI. COMPUT. Vol. 4, No., pp. 48 56 c Society for Industrial and Applied Mathematics A FOURTH-ORDER CENTRAL WENO SCHEME FOR MULTIDIMENSIONAL HYPERBOLIC SYSTEMS OF CONSERVATION LAWS DORON LEVY,
More informationA Central Compact-Reconstruction WENO Method for Hyperbolic Conservation Laws
A Central Compact-Reconstruction WENO Method for Hyperbolic Conservation Laws Kilian Cooley 1 Prof. James Baeder 2 1 Department of Mathematics, University of Maryland - College Park 2 Department of Aerospace
More informationFinite volumes for complex applications In this paper, we study finite-volume methods for balance laws. In particular, we focus on Godunov-type centra
Semi-discrete central schemes for balance laws. Application to the Broadwell model. Alexander Kurganov * *Department of Mathematics, Tulane University, 683 St. Charles Ave., New Orleans, LA 708, USA kurganov@math.tulane.edu
More informationCompact Central WENO Schemes for Multidimensional Conservation Laws
arxiv:math/9989v [math.na] 3 Nov 999 Compact Central WENO Schemes for Multidimensional Conservation Laws Doron Levy Gabriella Puppo Giovanni Russo Abstract We present a new third-order central scheme for
More informationOn the Reduction of Numerical Dissipation in Central-Upwind Schemes
COMMUNICATIONS IN COMPUTATIONAL PHYSICS Vol., No., pp. 4-63 Commun. Comput. Phys. February 7 On the Reduction of Numerical Dissipation in Central-Upwind Schemes Alexander Kurganov, and Chi-Tien Lin Department
More informationICES REPORT A Multilevel-WENO Technique for Solving Nonlinear Conservation Laws
ICES REPORT 7- August 7 A Multilevel-WENO Technique for Solving Nonlinear Conservation Laws by Todd Arbogast, Chieh-Sen Huang, and Xikai Zhao The Institute for Computational Engineering and Sciences The
More informationApplication of the Kurganov Levy semi-discrete numerical scheme to hyperbolic problems with nonlinear source terms
Future Generation Computer Systems () 65 7 Application of the Kurganov Levy semi-discrete numerical scheme to hyperbolic problems with nonlinear source terms R. Naidoo a,b, S. Baboolal b, a Department
More informationA class of the fourth order finite volume Hermite weighted essentially non-oscillatory schemes
Science in China Series A: Mathematics Aug., 008, Vol. 51, No. 8, 1549 1560 www.scichina.com math.scichina.com www.springerlink.com A class of the fourth order finite volume Hermite weighted essentially
More informationCENTRAL DISCONTINUOUS GALERKIN METHODS ON OVERLAPPING CELLS WITH A NON-OSCILLATORY HIERARCHICAL RECONSTRUCTION
CENTRAL DISCONTINUOUS GALERKIN METHODS ON OVERLAPPING CELLS WITH A NON-OSCILLATORY HIERARCHICAL RECONSTRUCTION YINGJIE LIU, CHI-WANG SHU, EITAN TADMOR, AND MENGPING ZHANG Abstract. The central scheme of
More informationHierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws
Hierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws Dedicated to Todd F. Dupont on the occasion of his 65th birthday Yingjie Liu, Chi-Wang Shu and Zhiliang
More informationYINGJIE LIU, CHI-WANG SHU, EITAN TADMOR, AND MENGPING ZHANG
CENTRAL DISCONTINUOUS GALERKIN METHODS ON OVERLAPPING CELLS WITH A NON-OSCILLATORY HIERARCHICAL RECONSTRUCTION YINGJIE LIU, CHI-WANG SHU, EITAN TADMOR, AND MENGPING ZHANG Abstract. The central scheme of
More informationAn Improved Non-linear Weights for Seventh-Order WENO Scheme
An Improved Non-linear Weights for Seventh-Order WENO Scheme arxiv:6.06755v [math.na] Nov 06 Samala Rathan, G Naga Raju Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur,
More informationConvex ENO Schemes for Hamilton-Jacobi Equations
Convex ENO Schemes for Hamilton-Jacobi Equations Chi-Tien Lin Dedicated to our friend, Xu-Dong Liu, notre Xu-Dong. Abstract. In one dimension, viscosit solutions of Hamilton-Jacobi (HJ equations can be
More informationHierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws
Hierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws Dedicated to Todd F. Dupont on the occasion of his 65th birthday Yingjie Liu, Chi-Wang Shu and Zhiliang
More informationc 2001 Society for Industrial and Applied Mathematics
SIAM J. SCI. COMPUT. Vol. 3, No. 3, pp. 707 740 c 001 Society for Industrial and Applied Mathematics SEMIDISCRETE CENTRAL-UPWIND SCHEMES FOR HYPERBOLIC CONSERVATION LAWS AND HAMILTON JACOBI EQUATIONS ALEXANDER
More informationA numerical study of SSP time integration methods for hyperbolic conservation laws
MATHEMATICAL COMMUNICATIONS 613 Math. Commun., Vol. 15, No., pp. 613-633 (010) A numerical study of SSP time integration methods for hyperbolic conservation laws Nelida Črnjarić Žic1,, Bojan Crnković 1
More informationA NUMERICAL STUDY FOR THE PERFORMANCE OF THE RUNGE-KUTTA FINITE DIFFERENCE METHOD BASED ON DIFFERENT NUMERICAL HAMILTONIANS
A NUMERICAL STUDY FOR THE PERFORMANCE OF THE RUNGE-KUTTA FINITE DIFFERENCE METHOD BASED ON DIFFERENT NUMERICAL HAMILTONIANS HASEENA AHMED AND HAILIANG LIU Abstract. High resolution finite difference methods
More informationA re-averaged WENO reconstruction and a third order CWENO scheme for hyperbolic conservation laws
A re-averaged WENO reconstruction and a third order CWENO scheme for hyperbolic conservation laws Chieh-Sen Huang a,, Todd Arbogast b,, Chen-Hui Hung c,3 a Department of Applied Mathematics and National
More informationSemi-Discrete Central-Upwind Schemes with Reduced Dissipation for Hamilton-Jacobi Equations
Semi-Discrete Central-Upwind Schemes with Reduced Dissipation for Hamilton-Jacobi Equations Steve Bryson, Aleander Kurganov, Doron Levy and Guergana Petrova Abstract We introduce a new family of Godunov-type
More informationFinite-Volume-Particle Methods for Models of Transport of Pollutant in Shallow Water
Journal of Scientific Computing ( 006) DOI: 0.007/s095-005-9060-x Finite-Volume-Particle Methods for Models of Transport of Pollutant in Shallow Water Alina Chertock, Alexander Kurganov, and Guergana Petrova
More informationOrder of Convergence of Second Order Schemes Based on the Minmod Limiter
Order of Convergence of Second Order Schemes Based on the Minmod Limiter Boan Popov and Ognian Trifonov July 5, 005 Abstract Many second order accurate non-oscillatory schemes are based on the Minmod limiter,
More informationWeighted Essentially Non-Oscillatory limiters for Runge-Kutta Discontinuous Galerkin Methods
Weighted Essentially Non-Oscillatory limiters for Runge-Kutta Discontinuous Galerkin Methods Jianxian Qiu School of Mathematical Science Xiamen University jxqiu@xmu.edu.cn http://ccam.xmu.edu.cn/teacher/jxqiu
More informationWell-Balanced Schemes for the Euler Equations with Gravity
Well-Balanced Schemes for the Euler Equations with Gravity Alina Chertock North Carolina State University chertock@math.ncsu.edu joint work with S. Cui, A. Kurganov, S.N. Özcan and E. Tadmor supported
More informationTHE ALTERNATING EVOLUTION METHODS FOR FIRST ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
THE ALTERNATING EVOLUTION METHODS FOR FIRST ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS HAILIANG LIU TO MARSHALL SLEMROD ON HIS 7TH BIRTHDAY WITH FRIENDSHIP AND APPRECIATION. Abstract. In this paper,
More informationImprovement of convergence to steady state solutions of Euler equations with. the WENO schemes. Abstract
Improvement of convergence to steady state solutions of Euler equations with the WENO schemes Shuhai Zhang, Shufen Jiang and Chi-Wang Shu 3 Abstract The convergence to steady state solutions of the Euler
More informationA Bound-Preserving Fourth Order Compact Finite Difference Scheme for Scalar Convection Diffusion Equations
A Bound-Preserving Fourth Order Compact Finite Difference Scheme for Scalar Convection Diffusion Equations Hao Li Math Dept, Purdue Univeristy Ocean University of China, December, 2017 Joint work with
More informationENO and WENO schemes. Further topics and time Integration
ENO and WENO schemes. Further topics and time Integration Tefa Kaisara CASA Seminar 29 November, 2006 Outline 1 Short review ENO/WENO 2 Further topics Subcell resolution Other building blocks 3 Time Integration
More informationSemi-discrete central-upwind schemes with reduced dissipation for Hamilton Jacobi equations
IMA Journal of Numerical Analysis 005 5, 113 138 doi: 10.1093/imanum/drh015 Semi-discrete central-upwind schemes with reduced dissipation for Hamilton Jacobi equations STEVE BRYSON Programme in Scientific
More informationALTERNATING EVOLUTION DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS
ALTERNATING EVOLUTION DISCONTINUOUS GALERKIN METHODS FOR HAMILTON-JACOBI EQUATIONS HAILIANG LIU AND MICHAEL POLLACK Abstract. In this work, we propose a high resolution Alternating Evolution Discontinuous
More informationEntropic Schemes for Conservation Laws
CONSTRUCTVE FUNCTON THEORY, Varna 2002 (B. Bojanov, Ed.), DARBA, Sofia, 2002, pp. 1-6. Entropic Schemes for Conservation Laws Bojan Popov A new class of Godunov-type numerical methods (called here entropic)
More informationDedicated to the 70th birthday of Professor Lin Qun
Journal of Computational Mathematics, Vol.4, No.3, 6, 39 5. ANTI-DIFFUSIVE FINITE DIFFERENCE WENO METHODS FOR SHALLOW WATER WITH TRANSPORT OF POLLUTANT ) Zhengfu Xu (Department of Mathematics, Pennsylvania
More informationA Finite Volume Code for 1D Gas Dynamics
A Finite Volume Code for 1D Gas Dynamics Michael Lavell Department of Applied Mathematics and Statistics 1 Introduction A finite volume code is constructed to solve conservative systems, such as Euler
More informationCentral Schemes for Systems of Balance Laws Salvatore Fabio Liotta, Vittorio Romano, Giovanni Russo Abstract. Several models in mathematical physics a
Central Schemes for Systems of Balance Laws Salvatore Fabio Liotta, Vittorio Romano, Giovanni Russo Abstract. Several models in mathematical physics are described by quasilinear hyperbolic systems with
More informationAnti-diffusive finite difference WENO methods for shallow water with. transport of pollutant
Anti-diffusive finite difference WENO methods for shallow water with transport of pollutant Zhengfu Xu 1 and Chi-Wang Shu 2 Dedicated to Professor Qun Lin on the occasion of his 70th birthday Abstract
More informationA parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows.
A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows Tao Xiong Jing-ei Qiu Zhengfu Xu 3 Abstract In Xu [] a class of
More informationPositivity-preserving high order schemes for convection dominated equations
Positivity-preserving high order schemes for convection dominated equations Chi-Wang Shu Division of Applied Mathematics Brown University Joint work with Xiangxiong Zhang; Yinhua Xia; Yulong Xing; Cheng
More informationA Speed-Up Strategy for Finite Volume WENO Schemes for Hyperbolic Conservation Laws
J Sci Comput (011) 46: 359 378 DOI 10.1007/s10915-010-9407-9 A Speed-Up Strategy for Finite Volume WENO Schemes for Hyperbolic Conservation Laws Fei Teng Li Yuan Tao Tang Received: 3 February 010 / Revised:
More informationOn Convergence of Minmod-Type Schemes
On Convergence of Minmod-Type Schemes Sergei Konyagin, Boan Popov, and Ognian Trifonov April 16, 2004 Abstract A class of non-oscillatory numerical methods for solving nonlinear scalar conservation laws
More informationAn Accurate Deterministic Projection Method for Hyperbolic Systems with Stiff Source Term
An Accurate Deterministic Projection Method for Hyperbolic Systems with Stiff Source Term Alexander Kurganov Department of Mathematics, Tulane University, 683 Saint Charles Avenue, New Orleans, LA 78,
More informationHigh Order Accurate Runge Kutta Nodal Discontinuous Galerkin Method for Numerical Solution of Linear Convection Equation
High Order Accurate Runge Kutta Nodal Discontinuous Galerkin Method for Numerical Solution of Linear Convection Equation Faheem Ahmed, Fareed Ahmed, Yongheng Guo, Yong Yang Abstract This paper deals with
More informationWeighted ENO Schemes
Xiaolei Chen Advisor: Prof. Xiaolin Li Department of Applied Mathematics and Statistics Stony Brook University, The State University of New York February 7, 014 1 3 Mapped WENO-Z Scheme 1D Scalar Hyperbolic
More informationNumerical Methods for Conservation Laws WPI, January 2006 C. Ringhofer C2 b 2
Numerical Methods for Conservation Laws WPI, January 2006 C. Ringhofer ringhofer@asu.edu, C2 b 2 2 h2 x u http://math.la.asu.edu/ chris Last update: Jan 24, 2006 1 LITERATURE 1. Numerical Methods for Conservation
More informationThird Order Reconstruction of the KP Scheme for Model of River Tinnelva
Modeling, Identification and Control, Vol. 38, No., 07, pp. 33 50, ISSN 890 38 Third Order Reconstruction of the KP Scheme for Model of River Tinnelva Susantha Dissanayake Roshan Sharma Bernt Lie Department
More informationc 2003 Society for Industrial and Applied Mathematics
SIAM J. SCI. COMPUT. Vol., No., pp. 57 7 c 00 Society for Industrial and Applied Mathematics ON THE ARTIFICIAL COMPRESSION METHOD FOR SECOND-ORDER NONOSCILLATORY CENTRAL DIFFERENCE SCHEMES FOR SYSTEMS
More informationAn Assessment of Semi-Discrete Central Schemes for Hyperbolic Conservation Laws
SANDIA REPORT SAND2003-3238 Unlimited Release Printed September 2003 An Assessment of Semi-Discrete Central Schemes for Hyperbolic Conservation Laws Mark A. Christon David I. Ketcheson Allen C. Robinson
More informationNUMERICAL SOLUTION OF HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS
NUMERICAL SOLUTION OF HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS JOHN A. TRANGENSTEIN Department of Mathematics, Duke University Durham, NC 27708-0320 Ш CAMBRIDGE ЩР UNIVERSITY PRESS Contents 1 Introduction
More informationInverse Lax-Wendroff Procedure for Numerical Boundary Conditions of. Conservation Laws 1. Abstract
Inverse Lax-Wendroff Procedure for Numerical Boundary Conditions of Conservation Laws Sirui Tan and Chi-Wang Shu 3 Abstract We develop a high order finite difference numerical boundary condition for solving
More informationBound-preserving high order schemes in computational fluid dynamics Chi-Wang Shu
Bound-preserving high order schemes in computational fluid dynamics Chi-Wang Shu Division of Applied Mathematics Brown University Outline Introduction Maximum-principle-preserving for scalar conservation
More informationFinite Volume Method
Finite Volume Method An Introduction Praveen. C CTFD Division National Aerospace Laboratories Bangalore 560 037 email: praveen@cfdlab.net April 7, 2006 Praveen. C (CTFD, NAL) FVM CMMACS 1 / 65 Outline
More informationRiemann Solvers and Numerical Methods for Fluid Dynamics
Eleuterio R Toro Riemann Solvers and Numerical Methods for Fluid Dynamics A Practical Introduction With 223 Figures Springer Table of Contents Preface V 1. The Equations of Fluid Dynamics 1 1.1 The Euler
More informationApplied Mathematics and Mechanics (English Edition) Transversal effects of high order numerical schemes for compressible fluid flows
Appl. Math. Mech. -Engl. Ed., 403), 343 354 09) Applied Mathematics and Mechanics English Edition) https://doi.org/0.007/s0483-09-444-6 Transversal effects of high order numerical schemes for compressible
More informationA central discontinuous Galerkin method for Hamilton-Jacobi equations
A central discontinuous Galerkin method for Hamilton-Jacobi equations Fengyan Li and Sergey Yakovlev Abstract In this paper, a central discontinuous Galerkin method is proposed to solve for the viscosity
More informationAn Eulerian-Lagrangian WENO Scheme for Nonlinear Conservation Laws
An Eulerian-Lagrangian WENO Scheme for Nonlinear Conservation Laws Chieh-Sen Huang a,, Todd Arbogast b, a Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 84, Taiwan, R.O.C.
More informationThe RAMSES code and related techniques I. Hydro solvers
The RAMSES code and related techniques I. Hydro solvers Outline - The Euler equations - Systems of conservation laws - The Riemann problem - The Godunov Method - Riemann solvers - 2D Godunov schemes -
More informationNon-Oscillatory Central Schemes for a Traffic Flow Model with Arrhenius Look-Ahead Dynamics
Non-Oscillatory Central Schemes for a Traffic Flow Model with Arrhenius Look-Ahead Dynamics Alexander Kurganov and Anthony Polizzi Abstract We develop non-oscillatory central schemes for a traffic flow
More informationThe one-dimensional equations for the fluid dynamics of a gas can be written in conservation form as follows:
Topic 7 Fluid Dynamics Lecture The Riemann Problem and Shock Tube Problem A simple one dimensional model of a gas was introduced by G.A. Sod, J. Computational Physics 7, 1 (1978), to test various algorithms
More informationRunge-Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter
Runge-Kutta discontinuous Galerkin method with a simple and compact Hermite WENO limiter Jun Zhu, inghui Zhong, Chi-Wang Shu 3 and Jianxian Qiu 4 Abstract In this paper, we propose a new type of weighted
More informationRESEARCH HIGHLIGHTS. WAF: Weighted Average Flux Method
RESEARCH HIGHLIGHTS (Last update: 3 rd April 2013) Here I briefly describe my contributions to research on numerical methods for hyperbolic balance laws that, in my view, have made an impact in the scientific
More informationA minimum entropy principle of high order schemes for gas dynamics. equations 1. Abstract
A minimum entropy principle of high order schemes for gas dynamics equations iangxiong Zhang and Chi-Wang Shu 3 Abstract The entropy solutions of the compressible Euler equations satisfy a minimum principle
More information30 crete maximum principle, which all imply the bound-preserving property. But most
3 4 7 8 9 3 4 7 A HIGH ORDER ACCURATE BOUND-PRESERVING COMPACT FINITE DIFFERENCE SCHEME FOR SCALAR CONVECTION DIFFUSION EQUATIONS HAO LI, SHUSEN XIE, AND XIANGXIONG ZHANG Abstract We show that the classical
More informationFinite difference Hermite WENO schemes for the Hamilton-Jacobi equations 1
Finite difference Hermite WENO schemes for the Hamilton-Jacobi equations Feng Zheng, Chi-Wang Shu 3 and Jianian Qiu 4 Abstract In this paper, a new type of finite difference Hermite weighted essentially
More informationA semi-lagrangian finite difference WENO scheme for scalar nonlinear conservation laws
A semi-lagrangian finite difference WENO scheme for scalar nonlinear conservation laws Chieh-Sen Huang a,, Todd Arbogast b,, Chen-Hui Hung c a Department of Applied Mathematics, National Sun Yat-sen University,
More informationA Fifth Order Flux Implicit WENO Method
A Fifth Order Flux Implicit WENO Method Sigal Gottlieb and Julia S. Mullen and Steven J. Ruuth April 3, 25 Keywords: implicit, weighted essentially non-oscillatory, time-discretizations. Abstract The weighted
More informationHigh order finite difference Hermite WENO schemes for the Hamilton-Jacobi equations on unstructured meshes
High order finite difference Hermite WENO schemes for the Hamilton-Jacobi equations on unstructured meshes Feng Zheng, Chi-Wang Shu and Jianxian Qiu 3 Abstract In this paper, a new type of high order Hermite
More informationFUNDAMENTALS OF LAX-WENDROFF TYPE APPROACH TO HYPERBOLIC PROBLEMS WITH DISCONTINUITIES
6th European Conference on Computational Mechanics (ECCM 6) 7th European Conference on Computational Fluid Dynamics (ECFD 7) 1115 June 018, Glasgow, UK FUNDAMENTALS OF LAX-WENDROFF TYPE APPROACH TO HYPERBOLIC
More informationMath 660-Lecture 23: Gudonov s method and some theories for FVM schemes
Math 660-Lecture 3: Gudonov s method and some theories for FVM schemes 1 The idea of FVM (You can refer to Chapter 4 in the book Finite volume methods for hyperbolic problems ) Consider the box [x 1/,
More informationLecture 16. Wednesday, May 25, φ t + V φ = 0 (1) In upwinding, we choose the discretization of the spatial derivative based on the sign of u.
Lecture 16 Wednesday, May 25, 2005 Consider the linear advection equation φ t + V φ = 0 (1) or in 1D φ t + uφ x = 0. In upwinding, we choose the discretization of the spatial derivative based on the sign
More informationPOSITIVITY PROPERTY OF SECOND-ORDER FLUX-SPLITTING SCHEMES FOR THE COMPRESSIBLE EULER EQUATIONS. Cheng Wang. Jian-Guo Liu
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS SERIES B Volume 3 Number May003 pp. 0 8 POSITIVITY PROPERTY OF SECOND-ORDER FLUX-SPLITTING SCHEMES FOR THE COMPRESSIBLE EULER EQUATIONS
More informationNumerical Solutions for Hyperbolic Systems of Conservation Laws: from Godunov Method to Adaptive Mesh Refinement
Numerical Solutions for Hyperbolic Systems of Conservation Laws: from Godunov Method to Adaptive Mesh Refinement Romain Teyssier CEA Saclay Romain Teyssier 1 Outline - Euler equations, MHD, waves, hyperbolic
More informationARTICLE IN PRESS Mathematical and Computer Modelling ( )
Mathematical and Computer Modelling Contents lists available at ScienceDirect Mathematical and Computer Modelling ournal homepage: wwwelseviercom/locate/mcm Total variation diminishing nonstandard finite
More informationFourier analysis for discontinuous Galerkin and related methods. Abstract
Fourier analysis for discontinuous Galerkin and related methods Mengping Zhang and Chi-Wang Shu Abstract In this paper we review a series of recent work on using a Fourier analysis technique to study the
More informationFinite volume schemes for multidimensional hyperbolic systems based on the use of bicharacteristics
Computing and Visualization in Science manuscript No. (will be inserted by the editor) Finite volume schemes for multidimensional hyperbolic systems based on the use of bicharacteristics Mária Lukáčová-Medvid
More informationA Central Discontinuous Galerkin Method for Hamilton-Jacobi Equations
J Sci Comput (00) 45: 404 48 DOI 0.007/s095-009-9340-y A Central Discontinuous Galerkin Method for Hamilton-Jacobi Equations Fengyan Li Sergey Yakovlev Received: 6 August 009 / Revised: November 009 /
More informationInternational Engineering Research Journal
Special Edition PGCON-MECH-7 Development of high resolution methods for solving D Euler equation Ms.Dipti A. Bendale, Dr.Prof. Jayant H. Bhangale and Dr.Prof. Milind P. Ray ϯ Mechanical Department, SavitribaiPhule
More informationHFVS: An Arbitrary High Order Flux Vector Splitting Method
HFVS: An Arbitrary High Order Flu Vector Splitting Method Yibing Chen, Song Jiang and Na Liu Institute of Applied Physics and Computational Mathematics, P.O. Bo 8009, Beijing 00088, P.R. China E-mail:
More informationHigh Resolution Methods for Time Dependent Problems with Piecewise Smooth Solutions
ICM 2002 Vol. Ill 747^757 High Resolution Methods for Time Dependent Problems with Piecewise Smooth Solutions Eitan Tadmor* Abstract A trademark of nonlinear, time-dependent, convection-dominated problems
More informationLinear high-resolution schemes for hyperbolic conservation laws: TVB numerical evidence 1
Linear high-resolution schemes for hyperbolic conservation laws: TVB numerical evidence C. Bona a,2, C. Bona-Casas a,3, J. Terradas b,4 arxiv:885v2 [gr-qc] 2 Dec 28 a Departament de Física. Universitat
More informationNumerische Mathematik
Numer. Math. ) :545 563 DOI.7/s--443-7 Numerische Mathematik A minimum entropy principle of high order schemes for gas dynamics equations iangxiong Zhang Chi-Wang Shu Received: 7 July / Revised: 5 September
More informationSimulation of the evolution of concentrated shear layers in a Maxwell fluid with a fast high-resolution finite-difference scheme
J. Non-Newtonian Fluid Mech. 84 (1999) 275±287 Simulation of the evolution of concentrated shear layers in a Maxwell fluid with a fast high-resolution finite-difference scheme Raz Kupferman 1,a,*, Morton
More informationGas Dynamics Equations: Computation
Title: Name: Affil./Addr.: Gas Dynamics Equations: Computation Gui-Qiang G. Chen Mathematical Institute, University of Oxford 24 29 St Giles, Oxford, OX1 3LB, United Kingdom Homepage: http://people.maths.ox.ac.uk/chengq/
More informationCentral schemes for conservation laws with application to shallow water equations
Central schemes for conservation laws with application to shallow water equations Giovanni Russo 1 Department of Mathematics and Computer Science University of Catania Viale A. Doria 6 95125 Catania Italy
More informationStability properties of a family of chock capturing methods for hyperbolic conservation laws
Proceedings of te 3rd IASME/WSEAS Int. Conf. on FLUID DYNAMICS & AERODYNAMICS, Corfu, Greece, August 0-, 005 (pp48-5) Stability properties of a family of cock capturing metods for yperbolic conservation
More informationArt Checklist. Journal Code: Article No: 6959
Art Checklist Journal Code: JCPH Article No: 6959 Disk Recd Disk Disk Usable Art # Y/N Format Y/N Remarks Fig. 1 Y PS Y Fig. 2 Y PS Y Fig. 3 Y PS Y Fig. 4 Y PS Y Fig. 5 Y PS Y Fig. 6 Y PS Y Fig. 7 Y PS
More informationON THE COMPARISON OF EVOLUTION GALERKIN AND DISCONTINUOUS GALERKIN SCHEMES
1 ON THE COMPARISON OF EVOLUTION GALERKIN AND DISCONTINUOUS GALERKIN SCHEMES K. BAUMBACH and M. LUKÁČOVÁ-MEDVIĎOVÁ Institute of Numerical Simulation, Hamburg University of Technology, 21 079 Hamburg, Germany,
More informationMAXIMUM PRINCIPLE AND CONVERGENCE OF CENTRAL SCHEMES BASED ON SLOPE LIMITERS
MATHEMATICS OF COMPUTATION S 005-5780054-7 Article electronically published on May 6, 0 MAXIMUM PRINCIPLE AND CONVERGENCE OF CENTRAL SCHEMES BASED ON SLOPE LIMITERS ORHAN MEHMETOGLU AND BOJAN POPOV Abstract.
More informationStrong Stability-Preserving (SSP) High-Order Time Discretization Methods
Strong Stability-Preserving (SSP) High-Order Time Discretization Methods Xinghui Zhong 12/09/ 2009 Outline 1 Introduction Why SSP methods Idea History/main reference 2 Explicit SSP Runge-Kutta Methods
More informationNon-linear Methods for Scalar Equations
Non-linear Methods for Scalar Equations Professor Dr. E F Toro Laboratory of Applied Mathematics University of Trento, Italy eleuterio.toro@unitn.it http://www.ing.unitn.it/toro October 3, 04 / 56 Abstract
More informationA method for avoiding the acoustic time step restriction in compressible flow
A method for avoiding the acoustic time step restriction in compressible flow Nipun Kwatra Jonathan Su Jón T. Grétarsson Ronald Fedkiw Stanford University, 353 Serra Mall Room 27, Stanford, CA 9435 Abstract
More informationCONSTRUCTION OF HIGH-ORDER ADAPTIVE IMPLICIT METHODS FOR RESERVOIR SIMULATION
CONSTRUCTION OF HIGH-ORDER ADAPTIVE IMPLICIT METHODS FOR RESERVOIR SIMULATION A REPORT SUBMITTED TO THE DEPARTMENT OF ENERGY RESOURCES ENGINEERING OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
More informationAN OPTIMALLY ACCURATE SPECTRAL VOLUME FORMULATION WITH SYMMETRY PRESERVATION
AN OPTIMALLY ACCURATE SPECTRAL VOLUME FORMULATION WITH SYMMETRY PRESERVATION Fareed Hussain Mangi*, Umair Ali Khan**, Intesab Hussain Sadhayo**, Rameez Akbar Talani***, Asif Ali Memon* ABSTRACT High order
More informationSolving the Euler Equations!
http://www.nd.edu/~gtryggva/cfd-course/! Solving the Euler Equations! Grétar Tryggvason! Spring 0! The Euler equations for D flow:! where! Define! Ideal Gas:! ρ ρu ρu + ρu + p = 0 t x ( / ) ρe ρu E + p
More informationRankine-Hugoniot-Riemann solver for steady multidimensional conservation laws with source terms
Rankine-Hugoniot-Riemann solver for steady multidimensional conservation laws with source terms Halvor Lund a,b,c,, Florian Müller a, Bernhard Müller b, Patrick Jenny a a Institute of Fluid Dynamics, ETH
More informationarxiv: v4 [physics.flu-dyn] 23 Mar 2019
High-Order Localized Dissipation Weighted Compact Nonlinear Scheme for Shock- and Interface-Capturing in Compressible Flows Man Long Wong a, Sanjiva K. Lele a,b arxiv:7.895v4 [physics.flu-dyn] 3 Mar 9
More informationNumerical Methods for Modern Traffic Flow Models. Alexander Kurganov
Numerical Methods for Modern Traffic Flow Models Alexander Kurganov Tulane University Mathematics Department www.math.tulane.edu/ kurganov joint work with Pierre Degond, Université Paul Sabatier, Toulouse
More informationA new type of Hermite WENO schemes for Hamilton-Jacobi equations
A new type of Hermite WENO schemes for Hamilton-Jacobi equations Jun Zhu and Jianxian Qiu Abstract In this paper, a new type of Hermite weighted essentially non-oscillatory (HWENO) schemes is designed
More information1. Introduction. We are interested in the scalar hyperbolic conservation law
SIAM J. NUMER. ANAL. Vol. 4, No. 5, pp. 1978 1997 c 005 Society for Industrial and Applied Mathematics ON CONVERGENCE OF MINMOD-TYPE SCHEMES SERGEI KONYAGIN, BOJAN POPOV, AND OGNIAN TRIFONOV Abstract.
More informationAn adaptive mesh redistribution method for nonlinear Hamilton-Jacobi equations in two- and three-dimensions
An adaptive mesh redistribution method for nonlinear Hamilton-Jacobi equations in two- and three-dimensions H.-Z. Tang, Tao Tang and Pingwen Zhang School of Mathematical Sciences, Peking University, Beijing
More information