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CENTRAL SCHEMES AND CONTACT DISCONTINUITIES

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Mathematical Modelling and Numerical Analysis ESAIM: M2ANMod´elisation Math´ematique et Analyse Num´erique Vol. 34, No6, 2000, pp. 1259–1275CENTRAL SCHEMES AND CONTACT DISCONTINUITIESAlexander Kurganov1and Guergana Petrova1Abstract. We introduce a family of new second-order Godunov-type central schemes for one-dimensio-nal systems of conservation laws. They are a less dissipative generalization of the central-upwindschemes, proposed in [A. Kurganov et al., submitted to SIAM J. Sci. Comput.], whose constructionis based on the maximal one-sided local speeds of propagation. We also present a recipe, which helpsto improve the resolution of contact waves. This is achieved by using the partial characteristic de-composition, suggested by Nessyahu and Tadmor [J. Comput. Phys. 87 (1990) 408–463], which isefficiently applied in the context of the new schemes. The method is tested on the one-dimensionalEuler equations, subject to different initial data, and the results are compared to the numerical solu-tions, computed by other second-order central schemes. The numerical experiments clearly illustratethe advantages of the proposed technique.Mathematics Subject Classification. 65M10, 65M05.Received: May 29, 2000. Revised: October 2, 2000.1. IntroductionIn this paper, we study second-order Godunov-type central schemes for one-dimensional systems of hyperbolicconservation lawsut+ f (u)x=0, (1.1)subject to the initial datau(x, 0) = u0(x). (1.2)Here, u(x, t):=(u1(x, t), ... ,uN(x, t)) is the unknown vector-function and f(u):=(f1(u), ... ,fN(u)) is theflux vector with the Jacobian A :=∂fi∂uj,i,j=1, ... ,N.A first-order stable central scheme was first introduced by Lax and Friedrichs in [5, 15]. Unfortunately, theLax-Friedrichs (LxF) scheme has a large numerical dissipation which leads to a poor resolution of the nonsmoothparts of the solution (unless a huge number of grid points is used, which may be practically impossible).Keywords and phrases. Euler equations of gas dynamics, partial characteristic decomposition, fully-discrete and semi-discretecentral schemes.1Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA. e-mail: [email protected],[email protected] EDP Sciences, SMAI 20001260 A. KURGANOV AND G. PETROVAIn [22], Nessyahu and Tadmor proposed a second-order generalization (the NT scheme) of the LxF scheme. TheNT scheme has smaller numerical viscosity and resolves shock waves, rarefactions and contact discontinuitiesmuch better than the first-order LxF scheme. Higher-order generalizations of the NT scheme were introducedin [3, 17, 21]. We refer the reader to [24, 25] for an alternative staggered approach, and to [1, 2, 9, 18, 19] forexamples of central schemes in the multidimensional case.A disadvantage of the staggered central schemes is that their numerical viscosity is proportional to (∆x)2r/∆t,where r is the formal order of the scheme. This results in excessive numerical dissipation if a small time stepis enforced or a long-time integration is performed. To overcome this difficulty, a new family of nonstaggeredfully- and semi-discrete central schemes has been recently introduced in [11–14]. The key idea behind theirconstruction is to obtain a more precise estimate of the width of the local Riemann fans by utilizing themaximal local speeds of propagation. This reduces the numerical viscosity to O(∆t)2r−1, and therefore makes itpossible to efficiently use central schemes when the time step is small.Even though the numerical dissipation of the aforementioned high-order central schemes is small, it is notsmall enough to provide a satisfactory resolution of contact discontinuities. A possible way to improve theresolution was proposed in [22]. The main idea is to isolate the less expensive characteristic projection onthe linear contact field, and to use the component-wise approach for the other, nonlinear fields (see Sect. 4in [22]). The effect of this partial characteristic decomposition can be amplified if it is used together with theartificial compression method (ACM), applied to the isolated contact field (see [6, 7, 22]). The above techniquewas extended in [20] in the multidimensional case. However, the drawback of using the ACM is that it requiresa certain a priori information about the solution in order to determine the optimal amount of the added“anti-diffusion”.The main purpose of this paper is to present (in the context of nonstaggered schemes from [13, 14]) analternative use of the partial characteristic information. First, in Section 2, we derive a new second-orderfully-discrete scheme and then, in Section 3, passing to the limit as ∆t → 0, we obtain a new second-ordersemi-discrete scheme. Compared with the schemes from [13, 14], the proposed schemes have smaller numericalviscosity, which leads to a sharper resolution of contact discontinuities.In Section 4, we use the partial characteristic decomposition to treat the linear and nonlinear fields separately,and this helps to improve the resolution of the contact waves. At the same time, the numerical examples,presented in Section 5, clearly demonstrate that the reduced numerical dissipation does not influence the stabilityof the proposed central schemes. We would also like to mention that the computational cost of the partialdecomposition, used in the construction of the new schemes, is relatively low. This preserves one of the mainadvantages of the central schemes – their efficiency.2. Fully-discrete schemeIn this section, we derive a modification of the second-order fully-discrete central-upwind scheme, introducedin [13]. First, we would like to recall the construction of central-upwind schemes (see [13]), which consist ofthree consecutive steps – reconstruction, evolution and projection back onto the original grid.We will consider only uniform grids and use the following notation, xj:= j∆x, xj±12:= (j ± 1/2)∆x,tn:= n∆t, λ := ∆t/∆x, unj≈ u(xj,tn), and¯unj≈ ¯u(xj,tn):=1∆xxj+12Zxj−12u(ξ,tn)dξ.Here, ∆x and ∆t are small spatial and time scales, respectively, and ¯unjare the computed cell averages at timelevel t = tn.CENTRAL SCHEMES AND CONTACT DISCONTINUITIES 1261ReconstructionAssume that at time level t = tnwe have computed the cell averages {¯unj}. We then reconstruct thesecond-order piecewise linear interpolanteu(x, tn)=lnj(x):=¯unj+ snj(x − xj),xj−12<x<xj+12, ∀j. (2.1)To ensure the non-oscillatory


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