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# Optimal L1-Rate of Convergence for The Viscosity Method and Monotone Scheme to Piecewise Constant Solutions with Shocks

SIAM Journal on Numerical Analysis, no. 3 (2006): 959-978

EI WOS SCOPUS

Abstract

We derive optimal error bounds for the viscosity method and monotone difference schemes to an initial-value problem of scalar conservation laws with initial data being a finite number of piecewise constants, subject to the initial discontinuities satisfying the entropy conditions. It is known that the entropy solution of the problem is pi...More

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Introduction

- Viscosity methods and monotone-difference schemes play an important role in both theoretical analysis and practical computation for hyperbolic conservation laws.
- Monotone-difference scheme, viscosity method, conservation laws, error estimate, convergence rate
- In this paper the authors justify this observation and derive optimal L1-error estimates for both the viscosity method and monotone difference scheme to piecewise constant solutions with a finite number of shock waves, i.e., solutions of the following initial-value problems: (1.1a)

Highlights

- Viscosity methods and monotone-difference schemes play an important role in both theoretical analysis and practical computation for hyperbolic conservation laws
- Hyman, and Lax [5] pointed out that the monotone-difference schemes are of at most first-order accuracy and Kuznetsov √[9, 10] showed that their L1-error bound for boundary value (BV) initial data is O ∆x as ∆x goes to zero, where ∆x is the size of space
- Tang and Teng [16] recently proved that all monotone schemes a√pplied to linear first-order equations with discontinuous i√nitial data is of at most ∆x rate of convergence in L1-norm. This means that the ∆x rate of convergence in L1-norm is the best possible for the monotone schemes applied to scalar conservation laws if it includes the linear case
- We will present some numerical experiments to verify the theoretical analysis, i.e., the convergence rates of monotone schemes to multishock wave solutions are of one order
- The numerical results clearly indicate that for the Lax–Friedrichs monotone-difference scheme, the convergence rates for both noninteracting and interacting shocks are of one order, which complies well with the theoretical analysis of Theorem 1.1

Results

- If v and w∆x are solutions of the viscosity method and monotone scheme, defined by (1.9) and (1.10) below, respectively, to the initial-value problem (1.1) and (1.2), the following uniform error bounds are fulfilled for any 0 < t < +∞: (1.7)
- W∆x(·, t) − u(·, t) L1(R) ≤ CK ∆x, where u is the entropy solution of (1.1) and (1.2), and ∆x are the viscosity coefficient and the discrete mesh length of the two approximate methods, respectively, and CK are constants independent of t, , and ∆x, but depend on K, the number of initial discontinuities.
- The rest of this paper is organized as follows: in section 2 the authors will introduce stability lemmas and travelling-wave lemmas for the viscosity method and monotone schemes.
- First the authors establish stability lemmas and travelling-wave lemmas for both the viscosity method and the monotone difference scheme.
- In what follows the authors introduce a travelling-wave solution to the monotone scheme
- With the aid of the stability estimate (2.4), the authors will prove that the L1-error of the difference between v (·, t) and v (·, t), the viscosity solution of (1.9), is bounded by O( ).
- With the aid of the stability estimate (2.16), the authors will prove that the L1error of the difference between w∆x(·, t) and w∆x(·, t), the monotone-scheme solution of (1.10), is bounded by O(∆x).
- The authors denote the finite-difference solution to the initial-value problem (1.9a) and (4.11) by w∆(1x) (x, t) and Theorem 1.1 shows that u(·, t + t(1)) − w∆(1x) (·, t) L1(R) ≤ CK−1 ∆x for 0 ≤ t < +∞.

Conclusion

- The authors will present some numerical experiments to verify the theoretical analysis, i.e., the convergence rates of monotone schemes to multishock wave solutions are of one order.
- The numerical results clearly indicate that for the Lax–Friedrichs monotone-difference scheme, the convergence rates for both noninteracting and interacting shocks are of one order, which complies well with the theoretical analysis of Theorem 1.1.
- In order to prove Theorem 1.1 for the more general case the authors need the following lemmas concerning the travelling-wave solutions.

- Table1: L1-errors and convergence rates for the Lax–Friedrichs scheme to (5.1) and (5.2)

Funding

- The research of the authors was supported by the National Natural Science Foundation of China and the Science Fund of the Education Commission of China

Reference

- M. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), pp. 1–21.
- B. Engquist and S. Osher, One-side difference approximations for nonlinear conservation laws, Math. Comp., 36 (1981), pp. 321-351.
- J. Goodman and Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Rational Mech. Anal., 121 (1992), pp. 235–265.
- A. Harten, The artificial compression method for computation of shocks and contact discontinuities, Comm. Pure Appl. Math., 30 (1977), pp. 611–638.
- A. Harten, J. M. Hyman, and P. D. Lax, On finite difference approximations and entropy conditions for shocks, Comm. Pure Appl. Math., 29 (1976), pp. 297–322.
- D. Hoff and J. Smoller, Error bounds for Glimm difference approximations for scalar conservation laws, Trans. Amer. Math. Soc., 289 (1985), pp. 611–642.
- G. Jennings, Discrete shocks, Comm. Pure Appl. Math., 27 (1979), pp. 25–37.
- S.N. Kruzkov, First order quasi-linear equations with several variables, Math. USSR-Sb., 10 (1970), pp. 217–243.
- N.N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, Comput. Math. Math. Phys., 16 (1976), pp. 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, pp. 183–197.
- P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1973.
- J. Liu and Z. Xin, L1-stability of stationary discrete shocks, Math. Comp., 60 (1993), pp. 233–244.
- J. Liu and Z. Xin, Nonlinear stability of discrete shocks for system of conservation laws, Arch. Rational Mech. Anal., 125 (1993), pp. 217–256.
- B.J. Lucier, Error bounds for the methods of Glimm, Godunov and LeVeque, SIAM J. Numer. Anal., 22 (1985), pp. 1074–1081.
- O.A. Oleinik, Uniqueness and stability of the generalized solution of the Cauchy problem for a quasilinear equation, Uspekhi Mat. Nauk, 2 (1959), pp. 165–170. Amer. Math. Soc.
- T. Tang and Z.H. Teng, The sharpness of Kuznetsov’s O( ∆x ) L1-error estimate for monotone difference schemes, Math. Comp., 64 (1995), pp. 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), pp. 110–127.
- T. Tang and Z.H. Teng, Viscosity methods for piecewise smooth solutions to scalar conservation laws, Math. Comp., 66 (1997), to appear.
- Z.H. Teng, On the accuracy of fractional step methods for conservation laws in twodimensions, SIAM J. Numer. Anal., 31 (1994), pp. 43–63.
- 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), pp. 401–424.

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