High order finite difference schemes for hyperbolic conservation laws on complex domains: Extrapolation and time integration techniques

• Autores: David Zorío Ventura
• Directores de la Tesis: Pep Mulet Mestre (dir. tes.) , Antonio Baeza Manzanares (dir. tes.)
• Lectura: En la Universitat de València ( España ) en 2016
• Idioma: inglés
• Número de páginas: 179
• Tribunal Calificador de la Tesis: Carlos Manuel Castro Barbero (presid.) , Rosa María Donat Beneito (secret.) , Armando Coco (voc.)
• Enlaces
  • Tesis en acceso abierto en: RODERIC
• Resumen

  • High-Resolution Shock-Capturing (HRSC) schemes constitute the state of the art for computing accurate numerical approximations to the solution of many hyperbolic systems of conservation laws, especially in computational fluid dynamics.

    In this context, the application of suitable numerical boundary conditions on domains with complex geometry has become a problem with certain difficulty that has been tackled in different ways according to the nature of the numerical methods and mesh type. In this work we present a new technique for the extrapolation of information from the interior of the computational domain to ghost cells designed for structured Cartesian meshes (which, as opposed to non-structured meshes, cannot be adapted to the morphology of the domain boundary).

    The aformentioned technique is based on the application of Lagrange interpolation equipped with detection of discontinuities that permits a data dependent extrapolation, with higher order at smooth regions and essentially non oscillatory properties near discontinuities.

    We also propose an alternative approach to develop a high order accurate scheme both in space and time, with the one that was proposed by Qiu and Shu for numerically solving hyperbolic conservation laws as starting point. Both methods are based on the conversion of time derivatives to spatial derivatives through the Cauchy-Kowalewski technique, following the Lax-Wendroff procedure. Such spatial derivatives are then discretized through the Shu-Osher finite difference procedure with an adequate upwind scheme. The alternative approach replaces the exact derivatives of the flux by approximations of the suitable order in order to reduce both the implementation and the computational cost, as well as a fluctuation control which avoids the expansion of large terms at the discretization of the high order derivatives.