Título: Stable high-order finite difference methods based on non-uniform grid point distributions
Ponente: Miguel Hermanns
Fecha: 30/01/2012 12:00h
Lugar: Sala Seminarios, Edificio Torretamarit
Title: Stable high-order finite difference methods based on non-uniform grid point distributions
Speaker: Miguel Hermanns
Date: 30/01/2012 12:00h
Location: Sala Seminarios, Edificio Torretamarit
Resumen
When solving large and complex fluid dynamic problems, the use of high-order finite difference methods leads to reduced computational costs and memory requirements. However, high-order finite difference methods tend to become unstable in the presence of boundaries and the imposition of boundary conditions. Different theoretical approaches exist to overcome these limitations, like the GKS theory or the summation-by-parts rule. Although all these approaches have shown to indeed lead to finite difference methods of orders q = 4 to 8, the extension to even higher orders seems to be too cumbersome or unclear.
In the present work a different approach is followed, namely the idea that the Runge phenomenon is behind the observed stability problems. By following the philosophy behind the Chebyshev interpolation theory, a non-uniform grid point distribution for piecewise polynomial interpolations of degree q < N is developed, being N+1 the number of grid points. The application of the developed methods to standard evolution problems like wave equation or convection-diffusion equation shows that the methods are stable for orders well beyond q = 10, recovering even the spectral accuracy in the limit q = N. Applications to stability analyses of fluid dynamic problems will also be presented, showing that the developed finite difference methods outperform in terms of accuracy and speed all the other existing finite difference methods and even the Chebyshev collocation method.
Breve Bio
El profesor Miguel Hermanns es Doctor en Ingeniería Aeroespacial por la Universidad Politécnica de Madrid (2006) y obtuvo el grado de Máster (2001) en Dinámica de Fluidos Experimental y Numérica en el “von Karman Institute for Fluid Dynamics” (Bélgica). Su trabajo investigador se ha centrado principalmente en modelización de procesos físicos que involucran fluidos e intercambios de energía, y en el desarrollo y aplicaciones de métodos numéricos de orden superior a todo tipo de problemas. En los últimos años también investiga en temas de eficiencia energética en edificios.
Abstract
When solving large and complex fluid dynamic problems, the use of high-order finite difference methods leads to reduced computational costs and memory requirements. However, high-order finite difference methods tend to become unstable in the presence of boundaries and the imposition of boundary conditions. Different theoretical approaches exist to overcome these limitations, like the GKS theory or the summation-by-parts rule. Although all these approaches have shown to indeed lead to finite difference methods of orders q = 4 to 8, the extension to even higher orders seems to be too cumbersome or unclear.
In the present work a different approach is followed, namely the idea that the Runge phenomenon is behind the observed stability problems. By following the philosophy behind the Chebyshev interpolation theory, a non-uniform grid point distribution for piecewise polynomial interpolations of degree q < N is developed, being N+1 the number of grid points. The application of the developed methods to standard evolution problems like wave equation or convection-diffusion equation shows that the methods are stable for orders well beyond q = 10, recovering even the spectral accuracy in the limit q = N. Applications to stability analyses of fluid dynamic problems will also be presented, showing that the developed finite difference methods outperform in terms of accuracy and speed all the other existing finite difference methods and even the Chebyshev collocation method.
Brief Bio
El profesor Miguel Hermanns es Doctor en Ingeniería Aeroespacial por la Universidad Politécnica de Madrid (2006) y obtuvo el grado de Máster (2001) en Dinámica de Fluidos Experimental y Numérica en el “von Karman Institute for Fluid Dynamics” (Bélgica). Su trabajo investigador se ha centrado principalmente en modelización de procesos físicos que involucran fluidos e intercambios de energía, y en el desarrollo y aplicaciones de métodos numéricos de orden superior a todo tipo de problemas. En los últimos años también investiga en temas de eficiencia energética en edificios.