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Numerical analysis for conservation laws using \(l_1\) minimization. (English) Zbl 1433.65176

The authors develop and analyze a new numerical scheme for solving hyperbolic conservation laws that combines the Lax Wendroff method with \(l_1\) regularization. The method developed here adds a new critical conservation constraint. The resulting method is proved to be equivalent to the well known lasso problem, guaranteeing both existence and uniqueness of the numerical solution. Consistency, convergence, and conservation of the suggested scheme are proved. It is shown that the scheme is TVD (Total Variation Diminishing) and it satisfies the weak entropy condition for conservation laws. Some numerical tests are presented to support the theoretical results.

MSC:

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65K10 Numerical optimization and variational techniques
35Q31 Euler equations

Software:

SPGL1; glmnet
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References:

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