Arc length based WENO scheme for Hamilton-Jacobi Equations

In this article, novel smoothness indicators are presented for calculating the nonlinear weights of weighted essentially non-oscillatory scheme to approximate the viscosity numerical solutions of Hamilton-Jacobi equations. These novel smoothness indicators are constructed from the derivatives of reconstructed polynomials over each sub-stencil. The constructed smoothness indicators measure the arc-length of the reconstructed polynomials so that the new nonlinear weights could get less absolute truncation error and gives a high-resolution numerical solution. Extensive numerical tests are conducted and presented to show the performance capability and the numerical accuracy of the proposed scheme with the comparison to the classical WENO scheme.Comment: 14 pages, 9 figure

Aproximación numérica de quinto orden de las ecuaciones de Hamilton-Jacobi

En este trabajo aproximamos la solución de viscosidad de las ecuaciones de Hamilton-Jacobi asociadas al problema de la reinicialización de curvas de nivel. Utilizamos para ello un método de quinto orden de precisión espacial óptimo para la aproximación de las ecuaciones de Hamilton-Jacobi. Como aplicación calculamos la aproximación a alto orden de funciones distancia euclidea signadas de curvas en R2

Esquemas de quinto orden Weighted Power‐ENO para ecuaciones de Hamilton‐Jacobi

En este trabajo han sido diseñados una clase de esquemas Weighted Power‐ENO empleados para aproximar la solución viscosa de las ecuaciones HJ. La idea esencial de este tipo de reconstrucción Power‐ENO es la aplicación a una clase de limitadores extendidos a diferencias de segundo orden en las reconstrucciones ENO clásicas de tercer orden, con el objetivo de mejorar las soluciones cerca de las discontinuidades. La estrategia de pesado basada en apropiados indicadores de suavidad lleva los esquemas a una precisión de quinto orden. Los experimentos numéricos realizados corroboran la precisión y la solidez de estos nuevos esquemas.Escuela Técnica superior de Ingeniería AgronómicaUniversidad Politécnica de Cartagen

Multi-Dimensional High Order Essentially Non-Oscillatory Finite Difference Methods in Generalized Coordinates

This project is about the development of high order, non-oscillatory type schemes for computational fluid dynamics. Algorithm analysis, implementation, and applications are performed. Collaborations with NASA scientists have been carried out to ensure that the research is relevant to NASA objectives. The combination of ENO finite difference method with spectral method in two space dimension is considered, jointly with Cai [3]. The resulting scheme behaves nicely for the two dimensional test problems with or without shocks. Jointly with Cai and Gottlieb, we have also considered one-sided filters for spectral approximations to discontinuous functions [2]. We proved theoretically the existence of filters to recover spectral accuracy up to the discontinuity. We also constructed such filters for practical calculations

Application of a 6DOF algorithm for the investigation of impulse waves generated due to sub-aerial landslides

Inland water bodies such as lakes, rivers and streams are generally considered safe from extreme wave events. Such inland water bodies are susceptible to extreme wave events due to impact of aerial landslides, where a large mass of land impacts the water at high velocities, resulting in a sudden transfer of momentum to the water body. Similar events can occur due to an underwater landslide as well. The evaluation of such extreme events in inland water bodies and the impact of such extreme waves on the regions adjacent to the water body is essential to assess the safety of the constructions on the banks of the water bodies. The generation of extreme waves due to aerial and sub-aerial landslides depends on several parameters such as the height of fall, the composition of the impacting land mass and the bottom slope of the water body. In this paper, the 6DOF algorithm implemented in the open source Computational Fluid Dy- namics (CFD) model REEF3D is used to simulate the motion of a sliding wedge impacting the water free surface. This is used to represent a sliding landmass impacting water after a landslide event. The wedge is represented using a primitive triangular surface mesh and a ray-tracing algorithm is used to determine the position of the object with respect to the underlying grid. Further, the level set method is then used to represent the solid boundary. The motion of the wedge is obtained by propagating the level set equation. The interaction of the wedge with the free water surface is obtained in a sharp and accurate manner using the level set method for both the water free surface and the solid boundary. REEF3D uses a staggered Cartesian numerical grid with a fifth-order WENO scheme for convection discretisation and a third-order Runge- Kutta scheme for time advancement. With the higher-order methods and the level set method, the model can be used to calculate detailed flow information such as the pressure changes in the water on impact and the associated deformation of the water free surface. The accurate representation of these characteristics is essential for correctly evaluating the height and period of the generated extreme wave and associated properties such as the wave celerity and wave run up on the banks during the extreme event

A WENO finite difference scheme for a new class of Hamilton-Jacobi equations in electroelastostatics

Hamilton-Jacobi equations have repeatedly emerged in many fields of physics, most notably, optimal control, differential games, geometric optics, and image processing. This thesis presents a new numerical method to solve a new class of Hamilton-Jacobi equation that has recently appeared in the context of nonlinear electroelastostatics. In a pioneering contribution, Crandall and Lions (1983) proved that a certain type of first-order finite difference method converges to the viscosity solution of a special class of Hamilton-Jacobi equations. From then on several successful methods of high-order approximation have been proposed in the literature, including the so-called WENO finite difference schemes. These schemes, however, were developed and tested for special classes of Hamilton-Jacobi equations, which do not include the general type of Hamilton-Jacobi equation of interest in this work. The objective of this thesis is to extend high-order WENO finite difference schemes to the most general type of Hamilton-Jacobi equations involving non-periodic boundary conditions in the "space" variables. Following its derivation, the proposed WENO scheme is tested for several cases involving one and two "space" variables for which there are analytical solutions available for arbitrarily large values of the "time" variable. These numerical experiments provide insight into the stability and rate of convergence of the method as "time" increases. They also provide insight into how errors propagate into the domain of computation due to non-periodic boundary conditions. This thesis concludes with the application of the method to compute the effective stored-energy function of an elastomer containing an isotropic distribution of vacuous pores under arbitrary 3D deformations

Fronts propagating with signal dependent speed in limited diffusion and related Hamilton-Jacobi formulations

We consider a class of limited diffusion equations and explore the formation of diffusion fronts as the result of a combination of diffusive and hyperbolic transport. We analyze a new class of Hamilton-Jacobi equations arising from the convective part of general Fokker-Planck equations ruled by a non-negative diffusion coefficient that depends on the unknown and on the gradient of the unknown. We explore the main features of the solution of the Hamilton-Jacobi equations that contain shocks and propose a suitable numerical scheme that approximates the solution in a consistent way with respect to the solution of the associated Fokker-Planck equation. We analyze three model problems covering different scenarios. One is the relativistic heat equation model where the speed of propagation of fronts is constant. A second one is a standard porous media model where the speed of propagation of fronts is a function of the density, is unbounded and can exceed any fixed value. We propose a third one which is a porous media model whose speed of propagating fronts depends on the density media and is limited. The three model problems satisfy a general Darcy law. We perform a set of numerical experiments under different piecewise smooth initial data with compact support and compare the behavior of the three different model problems