195 research outputs found

    Galerkin approximations for the optimal control of nonlinear delay differential equations

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    Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042

    Galerkin approximations for the optimal control of nonlinear delay differential equations

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    Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost function-als and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation

    An Accurate and Robust Numerical Scheme for Transport Equations

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    En esta tesis se presenta una nueva técnica de discretización para ecuaciones de transporte en problemas de convección-difusión para el rango completo de números de Péclet. La discretización emplea el flujo exacto de una ecuación de transporte unidimensional en estado estacionario para deducir una ecuación discreta de tres puntos en problemas unidimensionales y cinco puntos en problemas bidimensionales. Con "flujo exacto" se entiende que se puede obtener la solución exacta en función de integrales de algunos parámetros del fluido y flujo, incluso si estos parámetros son vari- ables en un volumen de control. Las cuadraturas de alto orden se utilizan para lograr resultados numéricos cercanos a la precisión de la máquina, incluso con mallas bastas.Como la discretización es esencialmente unidimensional, no está garantizada una solución con precisión de máquina para problemas multidimensionales, incluso en los casos en que las integrales a lo largo de cada coordenada cartesiana tienen una primitiva. En este sentido, la contribución principal de esta tesis consiste en una forma simple y elegante de obtener soluciones en problemas multidimensionales sin dejar de utilizar la formulación unidimensional. Además, si el problema es tal que la solución tiene precisión de máquina en el problema unidimensional a lo largo de las líneas coordenadas, también la tendrá para el dominio multidimensional.In this thesis, we present a novel discretization technique for transport equations in convection-diffusion problems across the whole range of Péclet numbers. The discretization employs the exact flux of a steady-state one-dimensional transport equation to derive a discrete equation with a three-point stencil in one-dimensional problems and a five-point stencil in two-dimensional ones. With "exact flux" it is meant that the exact solution can be obtained as a function of integrals of some fluid and flow parameters, even if these parameters are variable across a control volume. High-order quadratures are used to achieve numerical results close to machine- accuracy even with coarse grids. As the discretization is essentially one-dimensional, getting the machine- accurate solution of multidimensional problems is not guaranteed even in cases where the integrals along each Cartesian coordinate have a primitive. In this regard, the main contribution of this thesis consists in a simple and elegant way of getting solutions in multidimensional problems while still using the one-dimensional formulation. Moreover, if the problem is such that the solution is machine-accurate in the one-dimensional problem along coordinate lines, it will also be for the multidimensional domain.<br /

    Desarrollo de esquemas de muy alto orden con aplicación a flujos geofísicos

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    Los sistemas de ecuaciones en derivadas parciales de tipo hiperbólico se derivan de la aplicación de leyes de conservación a las magnitudes físicas fundamentales como la masa el momento o la energía y modelan una gran variedad de fenómenos físicos en el ámbito de la mecánica de fluidos. En este trabajo se estudia la resolución numérica de este tipo de sistemas ecuaciones mediante métodos numéricos desarrollados en el contexto de los volúmenes finitos. El trabajo se centra en esquemas numéricos de muy alto orden, que proporcionan una solución numérica más precisa y que resultan ser más eficientes conforme se refina la malla de cálculo. Se propone un método numérico de muy alto orden, denominado AR-ADER, de aplicación a sistemas de ecuaciones no lineales con términos fuentes. Además, se introducen una serie de mejoras en los procedimientos de reconstrucción WENO. Por otro lado, también se realiza la extensión a 2D de los procedimientos de reconstrucción WENO y de un esquema ADER para la resolución de ecuaciones lineales. El trabajo incluye resultados numéricos resultantes de la aplicación de los esquemas numéricos mencionados a diversos problemas que incluyen ecuaciones de transporte lineales, la ecuación de Burgers' y las ecuaciones de aguas poco profundas

    Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

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    This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

    Level Set Method for Simulating the Dynamics of the Fluid-Fluid Interfaces: Application of a Discontinuous Galerkin Method

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    A discontinuous Galerkin (DG) method was applied for simulating the dynamics of the fluid-fluid interfaces. The numerical implementations were performed in the context of an available in-house code, BoSSS. The flow field was assumed to be governed by a single-set of the Navier-Stokes equation in terms of the phase-dependent density and viscosity fields. As in the discontinuous Galerkin method the variables in each cell are expressed in terms of a polynomial space, the solution may exhibit spurious oscillations in the presence of the steep variations such as the density jumps across the interface. In order to overcome this problem, a diffuse interface assumption was made, according to which a jump is approximated by a continuous variation employing a regularized heaviside function. The interface diffusion is supposed to take place in a region with a reasonable width. Therefore, in order to properly express the smoothed jumps in terms of a polynomial space of a certain degree, only jumps with limited hight could be considered. Otherwise, the grid needs to be highly refined in the interface diffusion region for preventing the non-physical spatial oscillation of the solution. Surface tension effects as well as gravity were also involved in the simulations by adding the corresponding source terms to the Navier-Stokes equation. The interface kinematics was simulated using the level set method. Taking the advantage of the discontinuous Galerkin method, a precise solution to the level set advection equation was achieved. As the regularized Heaviside and delta functions are commonly expressed in terms of the level set function, the level set function needs to remain signed distance in order to keep a uniform diffusion width. The signed distance property of a level set functions was recovered by solving the re-initialization equation. A Godunov's scheme was applied for approximating the Hamiltonian of the re-initialization equation, in order to obtain a solution with a monotonicity preserving behavior. A notable stability improvement was achieved by adding an artificial diffusion along the characteristic lines of the re-initialization equation. The solution showed an appropriate hp-convergence behavior and almost no spurious movement of the interface was detected. For solving the Navier-Stokes equation, an explicit-implicit stiffly stable time integration method was employed combined by a splitting method for decoupling the velocity and pressure fields within the DG framework. This solver, which had been priorly implemented for the single phase formulation of the equation, was used as a basis for implementing a new solver for the multiphase formulation. The multiphase flow solver was verified by considering a number of the test cases, such as a rising bubble
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