High order accurate shock capturing schemes for hyperbolic conservation laws based on a new class of limiters.

Résumé

We have introduced new shock capturing schemes that reduce the numerical diusion at discontinuities, sharpen the discontinuities in derivative and avoid spurious oscillations, improving the behavior of essentially non oscillatory schemes and piecewise hyperbolic methods. We have introduced and analyzed in this work a new class of limiter functions, the so called power limiters", which are an essential tool for the construction of these schemes. When power limiters are used as limiters of rst or second order dierences, the resulting methods behave essentially non-oscillatory near discontinuities and they allow simple expressions of the local truncation errors when they are used as limiters of second order dierences. We have used the powereno limiter as a slope limiter for the design of a new piecewise hyperbolic method called the Power PHM method. The third order accurate Power PHM scheme improves the behavior of PHM at local extrema and contact discontinuities, and it shares the advantages of the PHM. Since these are compact schemes (three point stencil), Power PHM is recommended over PHM when this condition is convenient for the computation (e.g., relaxation schemes). We have also used the powereno limiter applied to consecutive second order nite dierences to construct the Power ENO method. We have analyzed a new fth order accurate Weighted Power ENO method as a nonlinear convex combination of the three Power ENO parabolas. Our fth order accurate Weighted PowerENO scheme improves the behavior of WENO5 reducing the numerical viscosity at contact discontinuities and local extrema. It captures ner scales for a xed computational grid. Our scheme is recommended when high order accuracy is a goal and when dealing with numerical schemes and simulations where a reduced (compact) stencil is not necessary. We have checked the robustness, stability and accuracy of the proposed schemes in a set of model problems by means of several numerical tests, including the shock entropy wave interaction, two interacting blast waves, the two dimensional four contacts Riemann problem and the two dimensional four shocks problem. Finally, we have shown the ability of the presented schemes in resolving ne scales near unstable interfaces by computing Rayleigh-Taylor and Richtmyer-Meshkov instabilities. ____________________________________________________________________________________________________ RESUMEN Hemos introducido nuevos metodos de captura de ondas de choque que reducen la difusion numerica en las discontinuidades, denen ntidamente las discontinuidades en derivada y evitan las oscilaciones espureas, mejorando el comportamiento de los esquemas esencialmente no oscilatorios y los metodos hiperbolicos a trozos. Hemos introducido y analizado en este trabajo una nueva clase de funciones limitadoras, los llamados power limiters" que son una herramienta esencial para la construccion de estos esquemas. Hemos utilizado el limitador powereno" como limitador de pendiente para el dise~no de un nuevo metodo hiperbolico a trozos que llamamos metodo Power PHM. Tambien hemos utilizado el limitador powereno aplicado a segundas diferencias contiguas para construir el metodo Power ENO. Hemos analizado un nuevo metodo de quinto orden de precisi on espacial, el metodo Weighted PowerENO, como una combinacion convexa no lineal de las tres parabolas PowerENO. Hemos comprobado la robustez, estabilidad y precision de los esquemas propuestos para un conjunto de problemas modelo mediante varios experimentos numericos. Finalmente hemos demostrado la capacidad de los esquemas presentados en la resolucion de las escalas nas en el entorno de interfases inestables mediante el calculo de inestabilidades de Rayleigh-Taylor y Richtmyer-Meshkov.