163 research outputs found
An explicit high-order single-stage single-step positivity-preserving finite difference WENO method for the compressible Euler equations
In this work we construct a high-order, single-stage, single-step
positivity-preserving method for the compressible Euler equations. Space is
discretized with the finite difference weighted essentially non-oscillatory
(WENO) method. Time is discretized through a Lax-Wendroff procedure that is
constructed from the Picard integral formulation (PIF) of the partial
differential equation. The method can be viewed as a modified flux approach,
where a linear combination of a low- and high-order flux defines the numerical
flux used for a single-step update. The coefficients of the linear combination
are constructed by solving a simple optimization problem at each time step. The
high-order flux itself is constructed through the use of Taylor series and the
Cauchy-Kowalewski procedure that incorporates higher-order terms. Numerical
results in one- and two-dimensions are presented
A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws
We introduce a new methodology for adding localized, space-time smooth,
artificial viscosity to nonlinear systems of conservation laws which propagate
shock waves, rarefactions, and contact discontinuities, which we call the
-method. We shall focus our attention on the compressible Euler equations in
one space dimension. The novel feature of our approach involves the coupling of
a linear scalar reaction-diffusion equation to our system of conservation laws,
whose solution is the coefficient to an additional (and artificial)
term added to the flux, which determines the location, localization, and
strength of the artificial viscosity. Near shock discontinuities, is
large and localized, and transitions smoothly in space-time to zero away from
discontinuities. Our approach is a provably convergent, spacetime-regularized
variant of the original idea of Richtmeyer and Von Neumann, and is provided at
the level of the PDE, thus allowing a host of numerical discretization schemes
to be employed. We demonstrate the effectiveness of the -method with three
different numerical implementations and apply these to a collection of
classical problems: the Sod shock-tube, the Osher-Shu shock-tube, the
Woodward-Colella blast wave and the Leblanc shock-tube. First, we use a
classical continuous finite-element implementation using second-order
discretization in both space and time, FEM-C. Second, we use a simplified WENO
scheme within our -method framework, WENO-C. Third, we use WENO with the
Lax-Friedrichs flux together with the -equation, and call this WENO-LF-C.
All three schemes yield higher-order discretization strategies, which provide
sharp shock resolution with minimal overshoot and noise, and compare well with
higher-order WENO schemes that employ approximate Riemann solvers,
outperforming them for the difficult Leblanc shock tube experiment.Comment: 34 pages, 27 figure
High-order Approximate Lax-Wendroff methods for systems of conservation laws
En esta tesis se introduce una nueva familia de métodos de alto orden: Los métodos Aproximadores Compactos Taylor para leyes de conservación (CAT por sus siglas en ingles). En estos métodos numéricos el proceso de Cauchy-Kovalevsky se evita al aplicar aproximaciones en forma recursiva. La diferencia a otros métodos aproximadores radica en que aquà se usan cómputos de los flujos numéricos de forma local, lo cual nos permite que los métodos tengan (2p+1) puntos en su esténcil y un orden de precisión 2p, donde p es un numero arbitrario entero. Aun mas, cuando el flujo es lineal estos métodos se reducen a los ya conocidos métodos de alto orden Lax-Wendroff y además son L2-estables table bajo la condición usual CFL. Sin embargo, los métodos CAT presentan un costo computacional extra por su carácter local, aunque este costo es compensado por el hecho que siguen dando buenos resultados aun con valores del CFL próximos a 1. Para evitar las oscilaciones que aparecen cerca de las discontinuidades se consideran aquà dos técnicas shock-capturing: la primera una nueva familia de métodos de alto orden, los métodos adaptativos compactos Taylor (ACAT), basados en la adaptación del orden del esquema acorde a una nueva familia de indicadores de suavidad.
La segunda técnica es la combinación de los métodos CAT con una variante original de los métodos WENO, nombrada : Approximate Taylor methods with fast and optimized weighted essentially nonoscillatory reconstructions (FOWENO-CAT)
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