Convergence, bounded variation properties and Kruzhkov solution of a fully discrete Lagrangian–Eulerian scheme via weak asymptotic analysis for 1D hyperbolic problems

• Published in: Numerical methods for partial differential equations Vol. 39; no. 3; pp. 2400 - 2443
• Main Authors: Abreu, Eduardo, Espírito Santo, Arthur, Lambert, Wanderson, Pérez, John
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 01.05.2023; Wiley Subscription Services, Inc
• Subjects: Asymptotic properties; Boundary value problems; Convergence; Eigenvalues; Entropy of solution; fully discrete Lagrangian–Eulerian scheme; Kruzhkov entropy condition; Maximum principle; no‐flow curves; positivity‐preserving; Rarefaction; Shock capturing; Shock waves; total variation nonincreasing; weak asymptotic analysis; weak bounded variation
• ISSN: 0749-159X; 1098-2426
• DOI: 10.1002/num.22972

We design and implement an effective fully discrete Lagrangian–Eulerian scheme for a class of scalar, local and nonlocal models, and systems of hyperbolic problems in 1D. We propose statements, via a weak asymptotic analysis, which include existence, uniqueness, regularity, and numerical approximations of entropy‐weak solutions computed with the scheme for the corresponding nonlinear initial value problem for the local scalar case. We study both convergence and weak bounded variation (BV) properties of the scheme to the entropy solution (for the local and scalar case) in the sense of Kruzhkov. The approach is based on the improved concept of no‐flow curves, as introduced by the authors, and we highlight the strengths of the method: (i) the scheme for systems of hyperbolic problems does not require computation of the eigenvalues (exact or approximate) either to the numerical flux function or the weak CFL stability condition (wCFL) and (ii) we prove the properties: positivity‐preserving, total variation nonincreasing, and maximum principle subject to the wCFL. We present numerical experiments to evaluate the shock capturing capabilities of the scheme in resolving the main features for hyperbolic problems: shock waves, rarefaction waves, contact discontinuities, positivity‐preserving properties, and nonlinear wave formations and interactions.