$L^1$-convergence rate of viscosity methods for scalar conservation laws with the interaction of elementary waves and the boundary
Authors:
Hongxia Liu and Tao Pan
Journal:
Quart. Appl. Math. 62 (2004), 601-621
MSC:
Primary 35L65; Secondary 65M15
DOI:
https://doi.org/10.1090/qam/2104264
MathSciNet review:
MR2104264
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Abstract: This paper is concerned with global error estimates for viscosity methods to initial-boundary problems for scalar conservation laws ${u_t} + f{\left ( u \right )_x} = 0$ on $\left [ {0, \infty } \right ) \times \left [ {0, \infty } \right )$, with the initial data $u\left ( {x, 0} \right ) = {u_0}\left ( x \right )$ and the boundary data $u\left ( {0, t} \right ) = {u_ - }$, where ${u_ - }$ is a constant, ${u_0}\left ( x \right )$ is a step function with a discontinuous point, and $f \in {C^2}$ satisfies $fâ > 0, f\left ( 0 \right ) = fâ\left ( 0 \right ) = 0$. The structure of global weak entropy solution of the inviscid problem in the sense of Bardos-Leroux-Nedelec [11] is clarified. If the inviscid solution includes the interaction that the central rarefaction wave collides with the boundary $x = 0$ and the boundary reflects a shock wave, then the error of the viscosity solution to the inviscid solution is bounded by $O\left ( {\epsilon ^{1/2}} + \epsilon \left | {In\epsilon } \right | + \epsilon \right )$ in ${L^1}$-norm. If the inviscid solution includes no interaction of the central rarefaction wave and the boundary or the interaction that the rarefaction wave collides with the boundary and is absorbed completely or partially by the boundary, then the error bound is $O\left ( \epsilon \left | {In\epsilon } \right | + \epsilon \right )$. In particular, if there is no central rarefaction wave included in the inviscid solution, the error bound is improved to $O\left ( \epsilon \right )$. The proof is given by a matching method and the traveling wave solutions.
- Jonathan Goodman and Zhou Ping Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Rational Mech. Anal. 121 (1992), no. 3, 235â265. MR 1188982, DOI https://doi.org/10.1007/BF00410614
- Zhen-Huan Teng and Pingwen Zhang, Optimal $L^1$-rate of convergence for the viscosity method and monotone scheme to piecewise constant solutions with shocks, SIAM J. Numer. Anal. 34 (1997), no. 3, 959â978. MR 1451109, DOI https://doi.org/10.1137/S0036142995268862
- David Hoff and Joel Smoller, Error bounds for Glimm difference approximations for scalar conservation laws, Trans. Amer. Math. Soc. 289 (1985), no. 2, 611â642. MR 784006, DOI https://doi.org/10.1090/S0002-9947-1985-0784006-5
N.N.Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, Compnt. Math. Math. Phys., 16(1976), 105-119.
- N. N. Kuznetsov, On stable methods for solving non-linear first order partial differential equations in the class of discontinuous functions, Topics in numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976) Academic Press, London, 1977, pp. 183â197. MR 0657786
- Bradley J. Lucier, Error bounds for the methods of Glimm, Godunov and LeVeque, SIAM J. Numer. Anal. 22 (1985), no. 6, 1074â1081. MR 811184, DOI https://doi.org/10.1137/0722064
- T. Tang and Zhen Huan Teng, The sharpness of Kuznetsovâs $O(\sqrt {\Delta x})\ L^1$-error estimate for monotone difference schemes, Math. Comp. 64 (1995), no. 210, 581â589. MR 1270625, DOI https://doi.org/10.1090/S0025-5718-1995-1270625-9
- T. Tang and Zhen Huan Teng, Error bounds for fractional step methods for conservation laws with source terms, SIAM J. Numer. Anal. 32 (1995), no. 1, 110â127. MR 1313707, DOI https://doi.org/10.1137/0732004
- Aslak Tveito and Ragnar Winther, An error estimate for a finite difference scheme approximating a hyperbolic system of conservation laws, SIAM J. Numer. Anal. 30 (1993), no. 2, 401â424. MR 1211397, DOI https://doi.org/10.1137/0730019
- T. Tang and Zhen-huan Teng, Viscosity methods for piecewise smooth solutions to scalar conservation laws, Math. Comp. 66 (1997), no. 218, 495â526. MR 1397446, DOI https://doi.org/10.1090/S0025-5718-97-00822-3
- C. Bardos, A. Y. le Roux, and J.-C. NĂ©dĂ©lec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations 4 (1979), no. 9, 1017â1034. MR 542510, DOI https://doi.org/10.1080/03605307908820117
- François Dubois and Philippe LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations 71 (1988), no. 1, 93â122. MR 922200, DOI https://doi.org/10.1016/0022-0396%2888%2990040-X
- K. T. Joseph and G. D. Veerappa Gowda, Explicit formula for the solution of convex conservation laws with boundary condition, Duke Math. J. 62 (1991), no. 2, 401â416. MR 1104530, DOI https://doi.org/10.1215/S0012-7094-91-06216-2
T.Pan and L.W.Lin, The global solution of the scalar nonconvex conservation law with boundary condition I; II, J. Part. Diff. Eqs., 8(1995), 371-383; 11(1998), 1-8.
H.X.Liu and T.Pan, The viscosity methods for Riemann initial-boundary problems of scalar conservation laws, Asian Information-Science-Life, 1(2)(2002), 133-144.
- Eberhard Hopf, On the right weak solution of the Cauchy problem for a quasilinear equation of first order, J. Math. Mech. 19 (1969/1970), 483â487. MR 0251357, DOI https://doi.org/10.1512/iumj.1970.19.19045
- Tung Chang and Ling Hsiao, The Riemann problem and interaction of waves in gas dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 41, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR 994414
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
O.A.Ladyzenskaja and N.N.Uralâceva, Boundary problems for linear and quasilinear parabolic equations, I-II, Izv. Akad. Nauk SSSR, ser. Mat. 26(1964), 2-52, 753-780 = AMS Transl. Ser.2, 47(1965), 217-299.
- Denis Serre and Kevin Zumbrun, Boundary layer stability in real vanishing viscosity limit, Comm. Math. Phys. 221 (2001), no. 2, 267â292. MR 1845324, DOI https://doi.org/10.1007/s002200100486
J.Goodman and Z.xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Rat. Mech. Anal., 121(1992), 235-265.
Z.H.Teng and P.W.Zhang, Optimal ${L^1}$-rate of convergence for viscosity method and monotone scheme to piecewise constant solutions with shocks, SIAM J. Numer. Anal., 34(1997), 959-978.
D.Hoff and J.Smoller, Error bounds for Glimm difference approximations for scalar conservation laws, Trans. Amer. Math. Soc., 289(1985), 611-642.
N.N.Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, Compnt. Math. Math. Phys., 16(1976), 105-119.
N.N.Kuznetsov, On stable methods for solving non-linear first order partial differential equations in the class of discontinuous functions, Topics in Numerical Analysis III, J. J. H. Miller, ed., Proc. Royal Irish Academy Conference, Academic Press, London, 1977, 183-197.
B.J.Lucier, Error bounds for the methods of Glimm, Godunov and Leveque, SIAM J. Numer. Anal., 22(1985), 1074-1081.
T.Tang and Z.H.Teng, The sharpness of Kuznetsovâs $O\left ( \sqrt {\Delta x} \right ){L^1}$-error estimate for monotone difference schemes, Math. Comp., 64(1995), 581-589.
T.Tang and Z.H.Teng, Error bounds for fractional step methods for conservation laws with source terms, SIAM J. Numer. Anal., 32(1995), 110-127.
A.Tveito and R.Winther, An error estimate for a finite difference scheme approximating a hyperbolic system of conservation laws, SIAM J. Numer. Anal., 30(1993), 401-424.
T.Tang and Z.H.Teng, Viscosity methods for piecewise smooth solutions to scalar conservation laws, Math. Comp., 66(1997), 495-526.
C.Bardos. A.Y.Leroux and J.C.Nedelc, First order quasilinear equations with boundary conditions, Comm. Part. Diff. Eqs., 4(1979), 1017-1034.
F.Dubois and P.LeFloch, Boundary conditions for nonlinear hyperboic systems of conservation laws, J. Diff. Eqs., 71(1988), 93-122.
K.T.Joseph and G.D.Veerappa Gowda, Explicit formula for the solution of convex conservation laws with boundary condition, Duke Math. J., 62(1991), 401-416.
T.Pan and L.W.Lin, The global solution of the scalar nonconvex conservation law with boundary condition I; II, J. Part. Diff. Eqs., 8(1995), 371-383; 11(1998), 1-8.
H.X.Liu and T.Pan, The viscosity methods for Riemann initial-boundary problems of scalar conservation laws, Asian Information-Science-Life, 1(2)(2002), 133-144.
E.Hopf, On the right weak solution of the Cauchy problem for a quasilinear equation of first order, J. Math. Mech., 19(1969), 483-487.
T.Chang and L.Hsiao, The Riemann problem and interaction of wave in gas dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol 41, Harlow, Longman Sci. Techn., 1989.
A.Friedman, Partial differential equations of parabolic type, Prentice Hall, New York, 1969.
O.A.Ladyzenskaja and N.N.Uralâceva, Boundary problems for linear and quasilinear parabolic equations, I-II, Izv. Akad. Nauk SSSR, ser. Mat. 26(1964), 2-52, 753-780 = AMS Transl. Ser.2, 47(1965), 217-299.
D.Serre and K.Zumbrun, Boundary layer stability in real vanishing viscosity limit, Commun. Math. Phys., 221(2001), 267-292.
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