ground state solutions for choquard equations with hardy littlewood sobolev upper critical growth and potential vanishing at infinity abstract in this paper we investigate the following non autonomous choquard equation u v x u i u 2 k x f u 2 u 2 2 u k x f u in r n where n 3 i is the riesz potential of order 0 n 2 n n 2 is the upper critical exponent due to the hardy littlewood sobolev inequality the functions v k c r n r may vanish at infinity and the function f c r r is noncritical and f t 0 t f s d s based on variational methods and the hardy type inequalities using the mountain pass theorem and the nehari manifold approach we prove the existence of ground state solution

an unconditionally stable fully conservative semi lagrangian method semi lagrangian methods have been around for some time dating back at least to 3 researchers have worked to increase their accuracy and these schemes have gained newfound interest with the recent widespread use of adaptive grids where the cfl based time step restriction of the smallest cell can be overwhelming since these schemes are based on characteristic tracing and interpolation they do not readily lend themselves to a fully conservative implementation however we propose a novel technique that applies a conservative limiter to the typical semi lagrangian interpolation step in order to guarantee that the amount of the conservative quantity does not increase during this advection in addition we propose a new second step that forward advects any of the conserved quantity that was not accounted for in the typical semi lagrangian advection we show that this new scheme can be used to conserve both mass and momentum for incompressible flows for incompressible flows we further explore properly conserving kinetic energy during the advection step but note that the divergence free projection results in a velocity field which is inconsistent with conservation of kinetic energy even for inviscid flows where it should be conserved for compressible flows we rely on a recently proposed splitting technique that eliminates the acoustic cfl time step restriction via an incompressible style pressure solve then our new method can be applied to conservatively advect mass momentum and total energy in order to exactly conserve these quantities and remove the remaining time step restriction based on fluid velocity that the original scheme still had

optimization based remap and transport a divide and conquer strategy for feature preserving discretizations this paper examines the application of optimization and control ideas to the formulation of feature preserving numerical methods with particular emphasis on the conservative and bound preserving remap constrained interpolation and transport advection of a single scalar quantity we present a general optimization framework for the preservation of physical properties and specialize it to a generic optimization based remap obr of mass density the latter casts remap as a quadratic program whose optimal solution minimizes the distance to a suitable target quantity subject to a system of linear inequality constraints the approximation of an exact mass update operator defines the target quantity which provides the best possible accuracy of the new masses without regard to any physical constraints such as conservation of mass or local bounds the latter are enforced by the system of linear inequalities in so doing the generic obr formulation separates accuracy considerations from the enforcement of physical properties we proceed to show how the generic obr formulation yields the recently introduced flux variable flux target fvft 1 and mass variable mass target mvmt 2 formulations of remap and then follow with a formal examination of their relationship using an intermediate flux variable mass target fvmt formulation we show the equivalence of fvft and mvmt optimal solutions to underscore the scope and the versatility of the generic obr formulation we introduce the notion of adaptable targets i e target quantities that reflect local solution properties extend fvft and mvmt to remap on the sphere and use obr to formulate adaptable conservative and bound preserving optimization based transport algorithms for cartesian and latitude longitude coordinate systems a selection of representative numerical examples on two dimensional grids demonstrates the computational properties of our approach

singularly perturbed critical choquard equations abstract in this paper we study the semiclassical limit for the singularly perturbed choquard equation e 2 u v x u e 3 r 3 q y g u y x y d y q x g u in r 3 where 0 3 e is a positive parameter v q are two continuous real function on r 3 and g is the primitive of g which is of critical growth due to the hardy littlewood sobolev inequality under suitable assumptions on g we first establish the existence of ground states for the critical choquard equation with constant coefficients next we establish existence and multiplicity of semi classical solutions and characterize the concentration behavior by variational methods

toward an oscillation free mass conservative eulerian lagrangian transport model ten numerical schemes to reduce spurious oscillations in transport simulations were implemented and tested in a eulerian lagrangian control volume finite element model the schemes included both new and existing flux corrected transport fct algorithms and nonlinear filters the ability of the methods to eliminate numerical oscillations while preserving mass and minimizing the introduction of numerical diffusion was compared in 2d tests of varying complexity the application of local mass correction algorithms was shown to be vital to avoid the introduction of mass errors by fct while none of the methods emerged as optimal for all cases a new fct method with a local mass correction scheme and a nonlinear filter can be recommended as the best approaches in general the method of choice depends on the problem being solved in particular on the concentration gradients and on the grid resolution

groundstates for a local nonlinear perturbation of the choquard equations with lower critical exponent abstract we prove the existence of ground state solutions by variational methods to the nonlinear choquard equations with a nonlinear perturbation u u i u n 1 u n 1 u f x u in r n where n 1 i is the riesz potential of order 0 n the exponent n 1 is critical with respect to the hardy littlewood sobolev inequality and the nonlinear perturbation f satisfies suitable growth and structural assumptions

multidimensional upwind methods for hyperbolic conservation laws abstract we present a class of second order conservative finite difference algorithms for solving numerically time dependent problems for hyperbolic conservation laws in several space variables these methods are upwind and multidimensional in that the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge and that all fluxes are evaluated and differenced at the same time in particular operator splitting is not used correct behavior at discontinuities is obtained by the use of solutions to the riemann problem and by limiting some of the second order terms numerical results are presented which show that the methods described here yield the same high resolution as the corresponding operator split methods

parallel processing for implicit solutions of the navier stokes equations an implicit scheme employing superposition of two three one dimensional solutions in two three dimensions is proposed for the solution of navier stokes equations is proposed boundary conditions are implemented in the discrete governing equations before the super position takes place hence no boundary conditions are needed for the one dimensional solutions details of the method as well as numerical confirmation of the analytical developments are discussed 9 refs

adaptive moment of fluid method a novel adaptive mesh refinement amr strategy based on the moment of fluid mof method for volume tracking of evolving interfaces is presented moment of fluid method is a new interface reconstruction and volume advection method using volume fractions as well as material centroids the mesh refinement criterion is based on the deviation of the actual centroid obtained by interface reconstruction from the reference centroid given by moment advection process the centroid error indicator detects not only high curvature regions but also regions with complicated subcell structures like filaments a new lagrange remap scheme is presented for advecting moments which includes lagrangian backtracking polygon intersection based remapping and forward tracking to define the material centroid the effectiveness and efficiency of amr mof method is demonstrated with classical test problems such as zalesak s disk and reversible vortex problem the comparison with previously published results for these problems shows the superior accuracy of the amr mof method over other methods in addition two new test cases with severe deformation rates are introduced namely droplet deformation and s shape deformation problems for further demonstration of the capabilities of the amr mof method

ground state solutions for a class of choquard equations with potential vanishing at infinity abstract this paper is dedicated to studying the following nonlinear choquard equation u v x u r n q y f u y x y d y q x f u u d 1 2 r n where n 3 0 n v c r n 0 q c r n 0 f c r r and f t 0 t f s d s by combining the non nehari manifold approach with some new inequalities we prove that the above equation has a ground state solution in the case when v x 0 as x where the strict monotonicity condition on f is not required moreover by using perturbation method we obtain the existence of a least energy solution in the zero mass case i e v 0 where f satisfies the condition that f t 0 0 for some t 0 r instead of the usual ambrosetti rabinowitz type condition these results extend the ones in alves figueiredo and yang 2015 1 and some related literature

ground state solutions for some indefinite variational problems abstract we consider the nonlinear stationary schrodinger equation u v x u f x u in r n here f is a superlinear subcritical nonlinearity and we mainly study the case where both v and f are periodic in x and 0 belongs to a spectral gap of v inspired by previous work of li et al 2006 11 and pankov 2005 13 we develop an approach to find ground state solutions i e nontrivial solutions with least possible energy the approach is based on a direct and simple reduction of the indefinite variational problem to a definite one and gives rise to a new minimax characterization of the corresponding critical value our method works for merely continuous nonlinearities f which are allowed to have weaker asymptotic growth than usually assumed for odd f we obtain infinitely many geometrically distinct solutions the approach also yields new existence and multiplicity results for the dirichlet problem for the same type of equations in a bounded domain

the repair paradigm new algorithms and applications to compressible flow the repair paradigm leads to several algorithms for redistributing mass momentum and energy while adhering to local maximum principles as an adjunct to the remapping step in certain compressible flow codes that use remapping such as arbitrary lagrangian eulerian codes or for just redistributing mass in advection codes in the case of advection of a concentration repair keeps the newly computed concentration in a cell between the maximum and minimum concentrations in neighboring old cells thus guaranteeing at least that the new concentration is between zero and one for compressible flow density velocity and internal energy are similarly constrained while maintaining conservation of mass momentum and total energy in this way positive density and internal energy are achieved as a side effect we propose a new algorithm combining both local and global repair that maintains causality and is efficient in a parallel computational setting the local global algorithm is independent of the order in which the distribution is performed and it maintains 1d symmetry this is applied to advection in two dimensions and to among others the leblanc problem the sedov problem and an interacting 2d blast wave problem the latter is done with a lagrangian code for which rezoning remapping and repair are essential

incremental remapping as a transport advection algorithm there are two fundamentally different strategies for solving the standard transport or continuity equation corresponding to whether it is expressed as a partial differential equation or as an integral statement of conservation the more common approach is to discretize the partial differential equation and to march the solution forward in time the alternative method is to project cell volumes along lagrangian trajectories as far forward or backward in time as desired and then to remap the resulting density distribution onto some target mesh this latter approach is known as remapping remapping has many advantages not the least of which is that the time step is limited only by accuracy considerations but it tends to be expensive and complex in this paper we show that if the time step is made sufficiently short such that trajectories are confined to the nearest neighbor cells then the remapping may be written as a flux form transport algorithm and it becomes nearly as simple and efficient as standard transport schemes the resulting method called incremental remapping retains most of the advantages of general remapping these include a geometric basis for transport b compatibility of associated tracer transport with simple tracer advection i e retention of tracer monotonicity properties and c efficient handling of multiple tracers since each additional tracer adds only a relatively small incremental cost

a subcell remapping method on staggered polygonal grids for arbitrary lagrangian eulerian methods we describe a new remapping algorithm for use in arbitrary lagrangian eulerian ale simulations the new features of this remapper are designed to complement a staggered mesh lagrangian phase in which the cells may be general polygons in two dimensions and which uses subcell discretizations to control unphysical mesh distortion and hourglassing our new remapping algorithm consists of three stages a gathering stage in which we interpolate momentum internal energy and kinetic energy to the subcells in a conservative way a subcell remapping stage in which we conservatively remap mass momentum internal and kinetic energy from the subcells of the lagrangian mesh to the subcells of the new rezoned mesh a scattering stage in which we conservatively recover the primary variables subcell density nodal velocity and cell centered specific internal energy on the new rezoned mesh we prove that our new remapping algorithm is conservative reversible and satisfies the debar consistency condition we also demonstrate computationally that our new remapping method is robust and accurate for a series of test problems in one and two dimensions

parallel processing scheme for the navier stokes equations part 1 scheme development using a lower and upper decomposition procedure an implicit parallel processing scheme for the compressible navier stokes equations is developed in this scheme the solution for a multidimensional problem is obtained by the superposition of two independent solution cycles because of the independence of these two solutions they can be performed on separate processors simultaneously to confirm the analytical developments numerical results are obtained using the proposed scheme and compared with other available solutions and experimental data a study of other characteristics of the proposed scheme such as stability symmetry preserving and effects of domain decomposition on convergence and stability is also conducted

a primitive variable method for the solution of three dimensional incompressible viscous flows abstract in this paper we present a new primitive variable method for the solution of the three dimensional incompressible reynolds averaged navier stokes equations in generalized curvilinear coordinates the governing equations are discretized on a non staggered grid and the discrete continuity equation is replaced by a discrete pressure poisson equation the discrete pressure equation is designed in such a way that i the compatibility condition for the poisson neumann problem is automatically satisfied and ii the discrete incompresibility constraint is satisfied to at least truncation error accuracy while the computed pressure is smooth the momentum equations are integrated in time using the four stage runge kutta algorithm while the pressure equation is solved using the point successive relaxation technique the method is applied to calculate the turbulent flow field over a ship model the computed results are in very good agreement with the experimental data

the repair paradigm and application to conservation laws repair is a conservative post processing procedure to be used in numerical methods for hyperbolic conservation laws in order to preserve certain qualitative characteristics of the numerical solution such as positivity of density and internal energy by means of redistribution of conserved quantities such as mass momentum and total energy among the cells of the mesh in this paper we describe the repair paradigm and prove several theorems which form a theoretical foundation for the repair procedures we consider two applications of repair and present corresponding numerical results the first application deals with improving properties of the remapping conservative interpolation stage of arbitrary lagrangian eulerian ale methods for the gas dynamics equations in which the solution is conservatively transferred from one mesh to another one requirement for remapping is that the interpolated density and internal energy on the new mesh have to stay positive another desirable property is that the remapping procedure should not create new extrema for the velocity field for various reasons these properties may not be satisfied especially for high order methods repair plays a supplemental role by bringing gas dynamics quantities into physically justified bounds another application of repair is to improve the quality of numerical methods for advection of some scalar tracer field with prescribed divergence free velocity field in which case the advection equation can be written as a conservation law and therefore the total amount of tracer is conserved we show how the repair procedure allows us to reduce oscillations in a numerical solution obtained by a formally high order method repair offers an alternative to more classical methods of reducing oscillations and maintaining positivity

formulation analysis and numerical study of an optimization based conservative interpolation remap of scalar fields for arbitrary lagrangian eulerian methods we develop and study the high order conservative and monotone optimization based remap obr of a scalar conserved quantity mass between two close meshes with the same connectivity the key idea is to phrase remap as a global inequality constrained optimization problem for mass fluxes between neighboring cells the objective is to minimize the discrepancy between these fluxes and the given high order target mass fluxes subject to constraints that enforce physically motivated bounds on the associated primitive variable density in so doing we separate accuracy considerations handled by the objective functional from the enforcement of physical bounds handled by the constraints the resulting obr formulation is applicable to general unstructured heterogeneous grids under some weak requirements on grid proximity but not on the cell types we prove that the obr algorithm is linearity preserving in one two and three dimensions the paper also examines connections between the obr and the recently proposed flux corrected remap fcr liska et al 1 we show that the fcr solution coincides with the solution of a modified version of obr m obr which has the same objective but a simpler set of box constraints derived by using a worst case scenario because m obr fcr has a smaller feasible set preservation of linearity may be lost and accuracy may suffer for some grid configurations our numerical studies confirm this and show that obr delivers significant increases in robustness and accuracy preliminary efficiency studies of obr reveal that it is only a factor of 2 1 slower than fcr but admits 1 5 times larger time steps

groundstates of nonlinear choquard equations existence qualitative properties and decay asymptotics we consider a semilinear elliptic problem delta u u i alpha ast abs u p abs u p 2 u quad text in r n where i alpha is a riesz potential and p 1 this family of equations includes the choquard or nonlinear schr odinger newton equation for an optimal range of parameters we prove the existence of a positive groundstate solution of the equation we also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point finally we derive the decay asymptotics at infinity of the groundstates

numerical solutions for the incompressible navier stokes equations in primitive variables using a non staggered grid 11 abstract in part i of this study a method was developed for the solution of the pressure poisson equation with neumann boundary conditions on a non staggered grid this method was used to determine the pressure when given the velocity field from a stream function vorticity solution in part ii the pressure equation is solved iteratively with the momentum navier stokes equations on a non staggered grid in this case the solution of the pressure equation not only provides the pressure but also serves to indirectly satisfy the continuity equation this primitive variables formulation has a major advantage over the stream function vorticity method in its applicability for three dimensional flow numerical results are obtained and compared with the stream function vorticity results for the driven cavity of part i

fast optimization based conservative remap of scalar fields through aggregate mass transfer abstract we develop a fast efficient and accurate optimization based algorithm for the high order conservative and local bound preserving remap constrained interpolation of a scalar conserved quantity between two close meshes with the same connectivity the new formulation is as robust and accurate as the flux variable flux target optimization based remap fvft obr 1 2 yet has the computational efficiency of an explicit remapper the coupled system of linear inequality constraints resulting from the flux form of remap is the main efficiency bottleneck in fvft obr while advection based remappers use the flux form to directly enforce mass conservation the optimization setting allows us to treat mass conservation as one of the constraints to take advantage of this fact we consider an alternative mass variable mass target mvmt obr formulation in which the optimization variables are the net mass updates per cell and a single linear constraint enforces the conservation of mass in so doing we change the structure of the obr problem from a global linear inequality constrained qp to a singly linearly constrained qp with simple bounds using the structure of the mvmt obr problem and the fact that in remap the old and new grids are close we are able to develop a simple efficient and easily parallelizable optimization algorithm for the primal mvmt obr qp numerical studies on a variety of affine and non affine grids confirm that mvmt obr is as accurate and robust as fvft obr but has the same computational cost as the explicit state of the art fcr

a subzone reconstruction algorithm for efficient staggered compatible remapping staggered grid lagrangian hydrodynamics algorithms frequently make use of subzonal discretization of state variables for the purposes of improved numerical accuracy generality to unstructured meshes and exact conservation of mass momentum and energy for arbitrary lagrangian eulerian ale methods using a geometric overlay it is difficult to remap subzonal variables in an accurate and efficient manner due to the number of subzone subzone intersections that must be computed this becomes prohibitive in the case of 3d unstructured polyhedral meshes a new procedure is outlined in this paper to avoid direct subzonal remapping the new algorithm reconstructs the spatial profile of a subzonal variable using remapped zonal and nodal representations of the data the reconstruction procedure is cast as an under constrained optimization problem enforcing conservation at each zone and node on the remapped mesh provides the set of equality constraints the objective function corresponds to a quadratic variation per subzone between the values to be reconstructed and a set of target reference values numerical results for various pure remapping and hydrodynamics tests are provided ideas for extending the algorithm to staggered grid radiation hydrodynamics are discussed as well as ideas for generalizing the algorithm to include inequality constraints

existence of groundstates for a class of nonlinear choquard equations we prove the existence of a nontrivial solution h n to the nonlinear choquard equation i in n where i is a riesz potential under almost necessary conditions on the nonlinearity in the spirit of berestycki and lions this solution is a groundstate if moreover is even and monotone on 0 then is of constant sign and radially symmetric

existence and concentration of ground state solutions for choquard equations involving critical growth and steep potential well abstract in the present paper we are interested in the following choquard type equation u v x u i u 2 u 2 2 u u p 2 u in r 3 where p 4 6 r r is a constant such that the operator l v x is non degenerate i is a riesz potential of order 0 3 2 3 is the upper critical exponent due to the hardy littlewood sobolev inequality the function v c r 3 r is nonnegative and has a potential well int v 1 0 additionally the operator has a sequence of dirichlet eigenvalues in h 0 1 expressed as 0 1 2 n n if 1 via the nehari manifold techniques and the ekeland variational principle we prove the existence and concentration of positive ground state solutions for sufficiently large if 1 and j for all j n employing the nehari pankov manifold methods and the constrained minimization arguments we obtain the existence of ground state solution for large enough and verify the asymptotic behaviour of ground state solutions as