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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

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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  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  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)
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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