Convergence Rate For The Ordered Upwind Method

The ordered upwind method (OUM) is used to approximate the viscosity solution of the static Hamilton---Jacobi---Bellman with direction-dependent weights on unstructured meshes. The method has been previously shown to provide a solution that converges to the exact solution, but no convergence rate has been theoretically proven. In this paper, it is shown that the solutions produced by the OUM in the boundary value formulation converge at a rate of at least the square root of the largest edge length in the mesh in terms of maximum error. An example with similar order of numerical convergence is provided.postprin

A Rotating-Grid Upwind Fast Sweeping Scheme for a Class of Hamilton-Jacobi Equations

We present a fast sweeping method for a class of Hamilton-Jacobi equations that arise from time-independent problems in optimal control theory. The basic method in two dimensions uses a four point stencil and is extremely simple to implement. We test our basic method against Eikonal equations in different norms, and then suggest a general method for rotating the grid and using additional approximations to the derivatives in different directions in order to more accurately capture characteristic flow. We display the utility of our method by applying it to relevant problems from engineering

Implicit upwind-Euler solution algorithms for unstructured-grid applications

The development of implicit upwind algorithms for the solution of the three-dimensional, time-dependent Euler equations on unstructured tetrahedral meshes is described. The implicit temporal discretization involves either a two-sweep Gauss-Seide relaxation procedure, a two-sweep Point-Jacobi relaxation procedure, or a single-sweep Point-Implicit procedure; the upwind spatial discretization is based on the flux-difference splitting of Roe. Detailed descriptions of the three implicit solution algorithms are given, and calculations for the Boeing 747 transport configuration are presented to demonstrate the algorithms. Advantages and disadvantages of the implicit algorithms are discussed. A steady-state solution for the 747 configuration, obtained at transonic flow conditions using a mesh of over 100,000 cells, required less than one hour of CPU time on a Cray-2 computer, thus demonstrating the speed and robustness of the general capability

Finite-Element Discretization of Static Hamilton-Jacobi Equations Based on a Local Variational Principle

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton-Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss-Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.Comment: 19 page

An efficient method for multiobjective optimal control and optimal control subject to integral constraints

We introduce a new and efficient numerical method for multicriterion optimal control and single criterion optimal control under integral constraints. The approach is based on extending the state space to include information on a "budget" remaining to satisfy each constraint; the augmented Hamilton-Jacobi-Bellman PDE is then solved numerically. The efficiency of our approach hinges on the causality in that PDE, i.e., the monotonicity of characteristic curves in one of the newly added dimensions. A semi-Lagrangian "marching" method is used to approximate the discontinuous viscosity solution efficiently. We compare this to a recently introduced "weighted sum" based algorithm for the same problem. We illustrate our method using examples from flight path planning and robotic navigation in the presence of friendly and adversarial observers.Comment: The final version accepted by J. Comp. Math. : 41 pages, 14 figures. Since the previous version: typos fixed, formatting improved, one mistake in bibliography correcte

Modeling transport of charged species in pore networks: solution of the Nernst-Planck equations coupled with fluid flow and charge conservation equations

A pore network modeling (PNM) framework for the simulation of transport of charged species, such as ions, in porous media is presented. It includes the Nernst-Planck (NP) equations for each charged species in the electrolytic solution in addition to a charge conservation equation which relates the species concentration to each other. Moreover, momentum and mass conservation equations are adopted and there solution allows for the calculation of the advective contribution to the transport in the NP equations. The proposed framework is developed by first deriving the numerical model equations (NMEs) corresponding to the partial differential equations (PDEs) based on several different time and space discretization schemes, which are compared to assess solutions accuracy. The derivation also considers various charge conservation scenarios, which also have pros and cons in terms of speed and accuracy. Ion transport problems in arbitrary pore networks were considered and solved using both PNM and finite element method (FEM) solvers. Comparisons showed an average deviation, in terms of ions concentration, between PNM and FEM below $5\%$ with the PNM simulations being over ${10}^{4}$ times faster than the FEM ones for a medium including about ${10}^{4}$ pores. The improved accuracy is achieved by utilizing more accurate discretization schemes for both the advective and migrative terms, adopted from the CFD literature. The NMEs were implemented within the open-source package OpenPNM based on the iterative Gummel algorithm with relaxation. This work presents a comprehensive approach to modeling charged species transport suitable for a wide range of applications from electrochemical devices to nanoparticle movement in the subsurface