Quinpi: Integrating Conservation Laws with CWENO Implicit Methods

Many interesting applications of hyperbolic systems of equations are stiff, and require the time step to satisfy restrictive stability conditions. One way to avoid small time steps is to use implicit time integration. Implicit integration is quite straightforward for first-order schemes. High order schemes instead also need to control spurious oscillations, which requires limiting in space and time also in the linear case. We propose a framework to simplify considerably the application of high order non-oscillatory schemes through the introduction of a low order implicit predictor, which is used both to set up the nonlinear weights of a standard high order space reconstruction, and to achieve limiting in time. In this preliminary work, we concentrate on the case of a third-order scheme, based on diagonally implicit Runge Kutta (DIRK) integration in time and central weighted essentially non-oscillatory (CWENO) reconstruction in space. The numerical tests involve linear and nonlinear scalar conservation laws

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 aquí 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 aquí 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)

Finite volume methods for unidirectional dispersive wave models

We extend the framework of the finite volume method to dispersive unidirectional water wave propagation in one space dimension. In particular we consider a KdV-BBM type equation. Explicit and IMEX Runge-Kutta type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariants conservation properties are also studied. Main applications include important nonlinear phenomena such as dispersive shock wave formation, solitary waves and their various interactions.Comment: 25 pages, 12 figures, 51 references. Other authors papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

A numerical study of SSP time integration methods for hyperbolic conservation laws

he method of lines approach for solving hyperbolic conservation laws is based on the idea of splitting the discretization process in two stages. First, the spatial discretization is performed by leaving the system continuous in time. This approximation is usually developed in a non-oscillatory manner with a satisfactory spatial accuracy. The obtained semi-discrete system of ordinary differential equations (ODE) is then solved by using some standard time integration method. In the last few years, a series of papers appeared, dealing with the high order strong stability preserving (SSP) time integration methods that maintain the total variation diminishing (TVD) property of the first order forward Euler method. In this work the optimal SSP Runge--Kutta methods of different order are considered in combination with the finite volume weighted essentially non-oscillatory (WENO) discretization. Furthermore, a new semi--implicit WENO scheme is presented and its properties in combination with different optimal implicit SSP Runge--Kutta methods are studied. Analysis is made on linear and nonlinear scalar equations and on Euler equations for gas dynamics

HIGH ORDER SHOCK CAPTURING SCHEMES FOR HYPERBOLIC CONSERVATION LAWS AND THE APPLICATION IN OPEN CHANNEL FLOWS

Many applications in engineering practice can be described by thehyperbolic partial differential equations (PDEs). Numerical modeling of this typeof equations often involves large gradients or shocks, which makes it achallenging task for conventional numerical methods to accurately simulate suchsystems. Thus developing accurate and efficient shock capturing numericalschemes becomes important for the study of hyperbolic equations.In this dissertation, a detailed study of the numerical methods for linearand nonlinear unsteady hyperbolic equations was carried out. A new finitedifference shock capturing scheme of finite volume style was developed. Thisscheme is based on the high order Pad?? type compact central finite differencemethod with the weighted essentially non-oscillatory (WENO) reconstruction toeliminate non-physical oscillations near the discontinuities while maintain stablesolution in the smooth areas. The unconditionally stable semi-implicit Crank-Nicolson (CN) scheme is used for time integration.The theoretical development was conducted based on one-dimensionalhomogeneous scalar equation and system equations. Discussions were alsoextended to include source terms and to deal with problems of higher dimension.For the treatment of source terms, Strang splitting was used. For multidimensionalequations, the ?? -form Douglas-Gunn alternating direction implicit(ADI) method was employed. To compare the performance of the scheme withENO type interpolation, the current numerical framework was also applied usingENO reconstruction.The numerical schemes were tested on 1-D and 2-D benchmark problems,as well as published experimental results. The simulated results show thecapability of the proposed scheme to resolve discontinuities while maintainingaccuracy in smooth regions. Comparisons with the experimental results validatethe method for dam break problems. It is concluded that the proposed scheme isa useful tool for solving hyperbolic equations in general, and from engineeringapplication perspective it provides a new way of modeling open channel flows

High Resolution Time-Limiter Schemes for Conservation Laws

This study investigates the high resolution time-limiter schemes for conservation laws. These schemes are proposed (K.Duraisamy, J.D.Baeder, J-G Liu, 2003) to enhance the stability of high order implicit time marching when the time step is beyond the original TVD limit. The improved stability is realized by taking local convex combination of a higher order oscillatory method (accurate mode) with a first order unconditionally TVD method (stable mode). The application of time-limiters, which detects the local smoothness, enables the self-adjusting switch between different modes. One of the main aspects of this work is employing time-limiters to improve the stability of the strongly S-stable DIRK3 scheme, which is shown to be non-SSP and thus may generate strong oscillations in non-smooth problems. The new Limited-DIRK3 scheme (L-DIRK3) is proposed. For convenience of applications to systems of equations, we also propose a new and convenient construction of time-limiter, which allows an arbitrary choice of reference quantity with minimal computation cost. Another innovation of our work is the extension of time-limiter schemes to multi-dimensional problems and convection-diffusion problems. The numerical results for one- and two-dimensional problems confirm that the L-DIRK3 scheme generate high resolution and less oscillatory solutions under large time step. Particularly, the L-DIRK3 scheme shows a clear improvement against the original DIRK3 in convection-diffusion problems when a large CFL number is taken