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Viscosity methods and monotone-difference schemes play an important role in both theoretical analysis and practical computation for hyperbolic conservation laws

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

Cited: 45|Views90

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optimal l1-rateinitial discontinuityentropy conditionentropy solutionerror boundMore(11+)

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.

Tables

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

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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

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Author

Zhen-huan Teng

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Pingwen Zhang (张平文)

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