a symmetric formulation for computing transient shallow water flows abstract the shallow water equations fall into the category of symmetric advective diffusive systems with source terms these types of equations can be very effectively solved using stabilized methods such as supg and gls a semi discrete finite element method based on these ingredients is presented for the computation of transient shallow water flows special care has been taken in the design of the operators in order to improve the performance for unsteady calculations the solution is advanced in time via a predictor multi corrector algorithm which includes as special cases the second order trapezoidal rule a first order explicit method and a first order implicit method equivalent to the constant in time element of the space time formulation

the velocity tracking problem for navier stokes flows with bounded distributed controls we present some systematic approaches to the mathematical analysis and numerical approximation of the time dependent optimal control problem of tracking the velocity for navier stokes flows in bounded two dimensional domains with bounded distributed controls we study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined we also define and analyze semidiscrete in time and fully space time discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system the results of some computational experiments are provided

optimal control problems for the navier stokes system coupled with the k ω turbulence model optimal control of fluid dynamics systems has gained attention in the last several years from the scientific community because of its potential use in design of new engineering devices and optimization of existing ones many research works have extensively studied the optimal control for the system of navier stokes but the problem of turbulence in these works is usually not taken into account because of the many difficulties arising from the numerical implementation and solution of the optimality system in this work turbulence is considered by coupling the k two equation turbulence model with the averaged navier stokes system the complete optimality system is derived and the existence of a weak solution proven some numerical examples are reported

a conservative formulation of the multidimensional upwind residual distribution schemes for general nonlinear conservation laws in the present paper we consider the numerical solution of systems of general nonlinear hyperbolic conservation laws on unstructured grids by means of the residual distribution method we propose a new formulation of the first order linear optimal positive n scheme relying on a contour integration of the convective fluxes over the boundaries of an element full conservation is achieved for arbitrary flux functions while the robustness and the monotone shock capturing of the original n scheme is retained the new variant of the n scheme is combined with the conservative second order linear lda scheme to obtain a nonlinear second order monotone b scheme the performance of the new residual distribution schemes is evaluated on problems governed by the euler equations as an application to a more complex system of conservation laws lacking an exact conservative linearization we solve the ideal magnetohydrodynamics equations in two spatial dimensions

constrained dirichlet boundary control in l 2 for a class of evolution equations optimal dirichlet boundary control based on the very weak solution of a parabolic state equation is analyzed this approach allows us to consider the boundary controls in l 2 which has advantages over approaches which consider control in sobolev spaces involving fractional derivatives pointwise constraints on the boundary are incorporated by the primal dual active set strategy its global and local superlinear convergences are shown a discretization based on space time finite elements is proposed and numerical examples are included

optimal controls of navier stokes equations this paper studies optimal control problems of the fluid flow governed by the navier stokes equations two control problems are formulated in the case of the driven cavity and flow through a channel with sudden expansion and solved successfully using a numerical optimization algorithm based on the augmented lagrangian method existence and the first order optimality condition of the optimal control are established a convergence result on the augmented lagrangian method for nonsmooth cost functional is obtained

solution of parabolic pseudo differential initial boundary value problems the motivation for the present work comes from a study of initialboundary value problems for the navier stokes equations g sl 2 33 these problems or rather their linearized versions are degenerate parabolic and we had found a method to get rid of the degeneracy by carrying the problems over to pseudo differential non degenerate parabolic problems where the results of the book gr2 could be used however the solvability statements for parabolic problems formulated in gr2 could be made more interesting and useful by some extra developments in particular a deeper analysis of the necessary compatibility conditions and an inclusion of all values of the sobolev space exponents also those that have an exceptional role in boundary value problems since this study has an interest not only for the navier stokes equations but also more generally e g whenever a reduction of a differential operator problem leads to a parabolic pseudo differential boundary value problem we decided to present it in a separate article we consider a problem

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

upwind residual discretization of enhanced boussinesq equations for wave propagation over complex bathymetries in this paper we consider the solution of the enhanced boussinesq equations of madsen and sorensen 1992 55 by means of residual based discretizations in particular we investigate the applicability of upwind and stabilized variants of continuous galerkin finite element and residual distribution schemes for the simulation of wave propagation and transformation over complex bathymetries these techniques have been successfully applied to the solution of the nonlinear shallow water equations see e g hauke 1998 39 and ricchiuto and bollermann 2009 61 in a first step toward the construction of a hybrid model coupling the enhanced boussinesq equations with the shallow water equations in breaking regions this paper shows that equal order and even low order second upwind stabilized techniques can be used to model non hydrostatic wave propagation over complex bathymetries this result is supported by theoretical truncation and dispersion error analyses and by thorough numerical validation

optimal control of vortices in non homogeneous navier stokes flows abstract optimal control of time dependent fluid flow governed by the incompressible navier stokes equations is considered a cost functional based on a local dynamical systems characterization of vortices is investigated the resulting functional is a non convex function of the velocity gradient tensor the optimality system based on a lagrangian formulation and adjoint equations describing first order necessary optimality conditions is provided the gradient and the second derivative of the cost functional with respect to the control are derived

stabilized residual distribution for shallow water simulations we propose a stabilized residual distribution rd scheme for the simulation of shallow water flows the final discretization is obtained combining the stabilized rd approach proposed in abgrall j comp phys 214 2006 and ricchiuto and abgrall iccfd4 springer verlag 2006 with the conservative formulation already used in ricchiuto et al j comp phys 222 2007 to simulate shallow water flows the scheme proposed is a nonlinear variant of a lax friedrichs type discretization it is well balanced it actually yields second order of accuracy in smooth areas and it preserves the positivity of the height of the water in presence of dry areas this is made possible by the residual character of the discretization by properly adapting the stabilization operators proposed in abgrall j comp phys 214 2006 and ricchiuto and abgrall iccfd4 springer verlag 2006 and thanks to the positivity preserving character of the underlying lax friedrichs scheme we demonstrate the properties of the discretization proposed on a wide variety of tests

pontryagin s maximum principle for optimal control of the stationary navier stokes equations this paper deals with pontryagin s maximum principle of the optimal control governed by stationary navier stokes equation some kind of state constraint is involved

the velocity tracking problem for wick stochastic navier stokes flows using weiner chaos expansion in this paper we numerically investigate a distributed control problem of stochastic incompressible newtonian fluid flows modeled by navier stokes equations with additive white noise wick product is applied to define the nonlinear convective term the objective of optimization is to track a given deterministic or stochastic velocity field an optimization algorithm based on weiner chaos expansion is proposed numerical tests are performed to validate our methodology

regularized state constrained boundary optimal control of the navier stokes equations abstract the numerical solution of the dirichlet boundary optimal control problem of the navier stokes equations in presence of pointwise state constraints is investigated two different regularization techniques are considered first a moreau yosida regularization of the problem is studied optimality conditions are derived and the convergence of the regularized solutions towards the original one is proved a source representation of the control combined with a lavrentiev type regularization strategy is also presented the analysis concerning optimality conditions and convergence of the regularized solutions is carried out in the last part of the paper numerical experiments are presented for the numerical solution of each regularized problem a semi smooth newton method is applied

an explicit residual based approach for shallow water flows we describe fully explicit residual based discretizations of the shallow water equations with friction on unstructured grids the schemes are obtained by properly adapting the explicit construction proposed in ricchiuto and abgrall 2010 57 in particular previous work on well balanced integration ricchiuto 2011 56 and preservation of the depth non negativity ricchiuto and bollermann 2009 60 is reformulated in the context of a genuinely explicit time stepping still based on a weighted residual approximation the paper discusses in depth how to achieve in this context an exact preservation of all the simple known steady equilibria and how to obtain a super consistent approximation for smooth non trivial moving equilibria the treatment of the wetting drying interface is also discussed giving formal conditions for the preservation of the non negativity of the depth for a particular case based on a nonlinear variant of a lax friedrichs type scheme the approach is analyzed and tested thoroughly the quality of the numerical results shows the interest in the proposed approach over previously proposed schemes in terms of accuracy and efficiency

analysis and approximation of the velocity tracking problem for navier stokes flows with distributed control we consider the mathematical formulation analysis and the numerical solution of a time dependent optimal control problem associated with the tracking of the velocity of a navier stokes flow in a bounded two dimensional domain through the adjustment of a distributed control the existence of optimal solutions is proved and the first order necessary conditions for optimality are used to derive an optimality system of partial differential equations whose solutions provide optimal states and controls semidiscrete in time and fully discrete space time approximations are defined and their convergence to the exact optimal solutions is shown a gradient method for the solution of the fully discrete equations is examined as are its convergence properties finally the results of some illustrative computational experiments are presented

unstructured finite volume discretisation of bed friction and convective flux in solute transport models linked to the shallow water equations the finite volume discretisation of the shallow water equations has been the subject of many previous studies most of which deal with a well balanced conservative discretisation of the convective flux and bathymetry however the bed friction discretisation has not been so profusely analysed in previous works while it may play a leading role in certain applications of shallow water models in this paper we analyse the numerical discretisation of the bed friction term in the two dimensional shallow water equations and we propose a new unstructured upwind finite volume discretisation for this term the new discretisation proposed improves the accuracy of the model in problems in which the bed friction is a relevant force in the momentum equation and it guarantees a perfect balance between gravity and bed friction under uniform flow conditions the relation between the numerical scheme used to solve the hydrodynamic equations and the scheme used to solve a scalar transport model linked to the shallow water equations is also analysed in the paper it is shown that the scheme used in the scalar transport model must take into consideration the scheme used to solve the hydrodynamic equations the most important implication is that a well balanced and conservative scheme for the scalar transport equation cannot be formulated just from the water depth and velocity fields but has to consider also the way in which the hydrodynamic equations have been solved

an edge based unstructured mesh discretisation in geospherical framework an arbitrary finite volume approach is developed for discretising partial differential equations governing fluid flows on the sphere unconventionally for unstructured mesh global models the governing equations are cast in the anholonomic geospherical framework established in computational meteorology the resulting discretisation retains proven properties of the geospherical formulation while it offers the flexibility of unstructured meshes in enabling irregular spatial resolution the latter allows for a global enhancement of the spatial resolution away from the polar regions as well as for a local mesh refinement a class of non oscillatory forward in time edge based solvers is developed and applied to numerical examples of three dimensional hydrostatic flows including shallow water benchmarks on a rotating sphere

high degree efficient symmetrical gaussian quadrature rules for the triangle gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method a brief review of existing quadrature rules for the triangle is given and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed new quadrature rules of degree 12 20 are presented and a short fortran program is included

solving inverse problems involving the navier stokes equations discretized by a lagrange galerkin method abstract in this article we are investigating the numerical approximation of an inverse problem involving the evolution of a newtonian viscous incompressible fluid described by the navier stokes equations in 2d this system is discretized using a low order finite element in space coupled with a lagrange galerkin scheme for the nonlinear advection operator we introduce a full discrete linearized scheme that is used to compute the gradient of a given cost function by ensuring its consistency using gradient based optimization algorithms we are able to deal with two fluid flow inverse problems the drag reduction around a moving cylinder and the identification of a far field velocity using the knowledge of the fluid load on a rectangular bluff body for both fixed and prescribed moving configurations

a well balanced scheme for the numerical processing of source terms in hyperbolic equations in a variety of physical problems one encounters source terms that are balanced by internal forces and this balance supports multiple steady state solutions that are stable typical of these are gravity driven flows such as those described by the shallow water equations over a nonuniform ocean bottom 1 10 qquad h t hu x 0 text and hu t left hu 2 frac gh 2 2 right x ga x x h 0 many classic numerical schemes cannot maintain these steady solutions or achieve them in the long time limit with an acceptable level of accuracy because they do not preserve the proper balance between the source terms and internal forces we propose here a numerical scheme adapted to a scalar conservation law that preserves this balance and that can it is hoped be extended to more general hyperbolic systems the proof of convergence of this scheme toward the entropy solution is given and some numerical tests are reported

a superconvergent hybridizable discontinuous galerkin method for dirichlet boundary control of elliptic pdes we begin an investigation of hybridizable discontinuous galerkin hdg methods for approximating the solution of dirichlet boundary control problems governed by elliptic pdes these problems can involve atypical variational formulations and often have solutions with low regularity on polyhedral domains these issues can provide challenges for numerical methods and the associated numerical analysis we propose an hdg method for a dirichlet boundary control problem for the poisson equation and obtain optimal a priori error estimates for the control specifically under certain assumptions for a 2d convex polygonal domain we show the control converges at a superlinear rate we present 2d and 3d numerical experiments to demonstrate our theoretical results

discontinuous galerkin approximations for an optimal control problem of three dimensional navier stokes voigt equations we analyze a fully discrete scheme based on the discontinuous in time galerkin approach which is combined with conforming finite element subspaces in space for the distributed optimal control problem of the three dimensional navier stokes voigt equations with a quadratic objective functional and box control constraints the space time error estimates of order o sqrt tau h where tau and h are respectively the time and space discretization parameters are proved for the difference between the locally optimal controls and their discrete approximations

entropy functions for symmetric systems of conservation laws abstract using a simple symmetrizability criterion we show that symmetric systems of conservation laws are equipped with a one parameter family of entropy functions

a semi smooth newton method for control constrained boundary optimal control of the navier stokes equations abstract in this paper we study optimal control of the navier stokes equations when the control acts as a pointwise constrained boundary condition of dirichlet type the problem is analyzed in the control space h 00 1 2 the optimality system and second order sufficient optimality conditions are derived for the numerical solution we apply a semi smooth newton method to a regularized version of the original problem and show convergence properties of the method and of the regularized solutions towards the original one

explicit runge kutta residual distribution schemes for time dependent problems second order case in this paper we construct spatially consistent explicit second order discretizations for time dependent hyperbolic problems starting from a given residual distribution rd discrete approximation of the steady operator we review the existing knowledge on consistent rd mass matrices and highlight the relations between different definitions we then introduce our explicit approach which is based on three main ingredients first recast the rd discretization as a stabilized galerkin scheme then use a shifted time discretization in the stabilization operator and lastly apply high order mass lumping on the galerkin component of the discretization the discussion is particularly relevant for schemes of the residual distribution type 18 3 which we will use for all our numerical experiments however similar ideas can be used in the context of residual based finite volume discretizations such as the ones proposed in 14 12 the schemes are tested on a wide variety of classical problems confirming the theoretical expectations

unsteady free surface flow simulation over complex topography with a multidimensional upwind technique in the context of numerical techniques for solving unsteady free surface problems finite element and finite volume approximations are widely used a class of upwind methods which attempts to model the equations in a genuinely multidimensional manner has been recently introduced as an alternative multidimensional upwind schemes mus were developed initially for the approximation of steady state solutions of the two dimensional euler equations on unstructured grids although they can be applicable to any system of hyperbolic conservation laws such as the shallow water equations the formal analogy between the two systems of equations is useful for simple cases however in practical applications of interest in hydraulics complex geometries and bottom slope variation can lead to important numerical errors produced by an inadequate source term discretization this problem has been analyzed and in this work the necessity of a multidimensional upwind discretization of the source terms is justified the basis of the numerical method is stated and the particular adaptation to unsteady shallow water flows over irregular geometry is described as test cases laboratory experimental data are used together with academic tests for validation

the velocity tracking problem for navier stokes flows with boundary control we present some systematic approaches to the mathematical formulation and numerical approximation of the time dependent optimal control problem of tracking the velocity for navier stokes flows in a bounded two dimensional domain with boundary control we study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined we also define and analyze semidiscrete in time and full space time discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system the results of some computational experiments are provided

optimal control for a stationary mhd system in velocity current formulation an optimal control problem for the equations governing the stationary problem of magnetohydrodynamics mhd is considered control mechanisms by external and injected currents and magnetic fields are treated an optimal control problem is formulated first order necessary and second order sufficient conditions are developed an operator splitting scheme for the numerical solution of the mhd state equations is analyzed

on positivity preserving finite volume schemes for euler equations we consider the positivity preserving property of first and higher order finite volume schemes for one and two dimensional euler equations of gas dynamics a general framework is established which shows the positivity of density and pressure whenever the underlying one dimensional first order building block based on an exact or approximate riemann solver and the reconstruction are both positivity preserving appropriate limitation to achieve a high order positivity preserving reconstruction is described

semidiscretization and error estimates for distributed control of the instationary navier stokes equations optimal control problems governed by the two dimensional instationary navier stokes equations and their spatial discretizations with finite elements are investigated a concept of semi discrete solutions to the control problem is introduced which is utilized to prove existence and uniqueness of discrete controls in neighborhoods of regular continuous solutions furthermore an optimal error estimate in terms of the spatial discretization parameter is given

constrained optimal control of navier stokes flow by semismooth newton methods abstract we propose and analyze a semismooth newton type method for the solution of a pointwise constrained optimal control problem governed by the time dependent incompressible navier stokes equations the method is based on a reformulation of the optimality system as an equivalent nonsmooth operator equation we analyze the flow control problem and prove q superlinear convergence of the method in the numerical implementation adjoint techniques are combined with a truncated conjugate gradient method numerical results are presented that support our theoretical results and confirm the viability of the approach

optimal control of fluid dynamic systems with state constraint of pointwise type abstract the aim of this work is to obtain the maximum principle by spike perturbation for the optimal control problem with pointwise type state constraint governed by 3 dimensional fluid dynamic systems

dynamics for controlled navier stokes systems with distributed controls the long time behavior of solutions for an optimal distributed control problem associated with the navier stokes equations is studied first a linear feedback solution for the navier stokes equations is constructed this feedback solution possesses decay in time properties then some preliminary estimates for the long time behavior of all solutions of the navier stokes equations are derived next the existence of a solution for the optimal control problem is proved finally the long time decay properties for the optimal solutions are established

error estimates for discontinuous galerkin time stepping schemes for robin boundary control problems constrained to parabolic pdes we consider fully discrete finite element approximations of a robin optimal boundary control problem constrained by linear parabolic pdes with rough initial data conforming finite element methods for spatial discretization combined with discontinuous time stepping galerkin schemes are being used for the space time discretization error estimates are proved under weak regularity hypotheses for the state adjoint and control variables computational examples validating our expected rates of convergence are also provided

on the rayleigh benard marangoni system and a related optimal control problem abstract in this paper we study a boundary control problem associated to the stationary rayleigh benard marangoni rbm system in presence of controls for the velocity and the temperature on parts of the boundary we analyze the existence uniqueness and regularity of weak solutions for the stationary rbm system in polyhedral domains of r 3 and then we prove the existence of the optimal solution by using the theorem of lagrange multipliers we derive an optimality system we also give a second order sufficient optimality condition and we establish a result of uniqueness of the optimal solution

boundary value problems and optimal boundary control for the navier stokes system the two dimensional case we study optimal boundary control problems for the two dimensional navier stokes equations in an unbounded domain control is effected through the dirichlet boundary condition and is sought in a subset of the trace space of velocity fields with minimal regularity satisfying the energy estimates an objective of interest is the drag functional we first establish three important results for inhomogeneous boundary value problems for the navier stokes equations namely we identify the trace space for the velocity fields possessing finite energy we prove the existence of a solution for the navier stokes equations with boundary data belonging to the trace space and we identify the space in which the stress vector along the boundary of admissible solutions is well defined then we prove the existence of an optimal solution over the control set finally we justify the use of lagrange multiplier principles derive an optimality system of equations in the weak sense from which optimal states and controls may be determined and prove that the optimality system of equations satisfies in appropriate senses a system of partial differential equations with boundary values

on the thermodynamics stability and hierarchy of entropy functions in fluid flow symmetric forms of systems of equations describing physical phenomena present exceptional desirable properties such as symmetric coefficient matrices well posedness of the cauchy initial value problem finite wave propagation speeds and a link to nonlinear stability principles therefore they should be used when possible to design better numerical methods in this paper such symmetric forms are reviewed for various models of fluid flow and a hierarchy of generalized entropy functions is presented standard concepts are extended to the diffusive terms

numerical solution of a flow control problem vorticity reduction by dynamic boundary action in order to laminarize an unsteady internal flow the vorticity field is minimized in a least squares sense using an optimal control approach the flow model is the navier stokes equation for a viscous incompressible fluid and the flow is controlled by suction and blowing on a part of the boundary a quasi newton method is used for the minimization of a quadratic objective function involving a measure of the vorticity and a regularization term the navier stokes equations are approximated using a finite difference scheme in time and finite element approximations in space accurate expressions for the gradient of the discrete objective function are needed to obtain a satisfactory convergence rate of the minimization algorithm therefore first order necessary conditions for a minimizer of the objective function are derived in the fully discrete case a memory saving device is discussed without which problems of any realistic size especially in three space dimensions would remain computationally intractable the feasibility of the optimal control approach for flow control problems is demonstrated by numerical experiments for a two dimensional flow in a rectangular cavity at a reynolds number high enough for nonlinear effects to be important