Advances in Numerical Methods for Hyperbolic Conservation Laws and Nonlinear Systems

The field of numerical methods for hyperbolic conservation laws and nonlinear systems is witnessing significant developments, with a focus on improving the accuracy and stability of numerical schemes. Researchers are exploring new approaches to solve random hyperbolic conservation laws, including the use of linear programming and measure-valued solutions. Additionally, there is a growing interest in finite-time stabilization of nonlinear systems, with a emphasis on designing feedback control laws that ensure convergence and rejection of perturbations. The development of new numerical methods, such as matrix-free convex limiting frameworks and high-order schemes, is also underway. These advancements have the potential to improve the resolution of discontinuities and enhance the overall performance of numerical simulations. Noteworthy papers include: High-Order Schemes for Hyperbolic Conservation Laws Using Young Measures, which proposes high-order extensions of existing schemes using linear programming, and A matrix-free convex limiting framework for continuous Galerkin methods with nonlinear stabilization, which introduces a new methodology for enforcing preservation of invariant domains.

Sources

High-Order Schemes for Hyperbolic Conservation Laws Using Young Measures

Finite-Time Stabilization of a Class of Nonlinear Systems in Hilbert Space

A matrix-free convex limiting framework for continuous Galerkin methods with nonlinear stabilization

A class of regularization schemes for linear ill-posed problems in Banach spaces under low order source conditions

Accuracy analysis and optimization of scale-independent third-order WENO-Z scheme with critical-point accuracy preservation

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