transient fokker planck kolmogorov equation solved with smoothed particle hydrodynamics method summary probabilistic theories aim at describing the properties of systems subjected to random excitations by means of statistical characteristics such as the probability density function pdf the time evolution of the pdf of the response of a randomly excited deterministic system is commonly described with the transient fokker planck kolmogorov fpk equation the fpk equation is a conservation equation of a hypothetical or abstract fluid which models the transport of probability this paper presents a generalized formalism for the resolution of the transient fpk equation by using the well known mesh free lagrangian method smoothed particle hydrodynamics numerical implementation shows notable advantages of this method in an unbounded state space 1 the conservation of total probability in the state space is explicitly written 2 no artifact is required to manage far field boundary conditions 3 the positivity of the pdf is ensured and 4 the extension to higher dimensions is straightforward furthermore thanks to the moving particles this method is adapted for a large kind of initial conditions even slightly dispersed distributions the fpk equation is solved without any a priori knowledge of the stationary distribution just a precise representation of the initial distribution is required copyright 2013 john wiley sons ltd

power penalty method for solving hjb equations arising from finance nonlinear hamilton jacobi bellman hjb equation commonly occurs in financial modeling implicit numerical scheme is usually applied to the discretization of the continuous hjb so as to find its numerical solution since it is generally difficult to obtain its analytic viscosity solution this type of discretization results in a nonlinear discrete hjb equation we propose a power penalty method to approximate this discrete equation by a nonlinear algebraic equation containing a power penalty term under some mild conditions we give the unique solvability of the penalized equation and show its convergence to the original discrete hjb equation moreover we establish a sharp convergence rate of the power penalty method which is of an exponential order with respect to the power of the penalty term we further develop a damped newton algorithm to iteratively solve the lower order penalized equation finally we present a numerical experiment solving an incomplete market optimal investment problem to demonstrate the rates of convergence and effectiveness of the new method we also numerically verify the efficiency of the power penalty method by comparing it with the widely used policy iteration method

improved incompressible smoothed particle hydrodynamics method for simulating flow around bluff bodies in this article we present numerical solutions for flow over an airfoil and a square obstacle using incompressible smoothed particle hydrodynamics isph method with an improved solid boundary treatment approach referred to as the multiple boundary tangents mbt method it was shown that the mbt boundary treatment technique is very effective for tackling boundaries of complex shapes also we have proposed the usage of the repulsive component of the lennard jones potential ljp in the advection equation to repair particle fractures occurring in the sph method due to the tendency of sph particles to follow the stream line trajectory this approach is named as the artificial particle displacement method numerical results suggest that the improved isph method which is consisting of the mbt method artificial particle displacement and the corrective sph discretization scheme enables one to obtain very stable and robust sph simulations the square obstacle and naca airfoil geometry with the angle of attacks between 0 and 15 were simulated in a laminar flow field with relatively high reynolds numbers we illustrated that the improved isph method is able to capture the complex physics of bluff body flows naturally such as the flow separation wake formation at the trailing edge and the vortex shedding the sph results are validated with a mesh dependent finite element method fem and excellent agreements among the results were observed

an arbitrary boundary with ghost particles incorporated in coupled fem sph model for fsi problems abstract it is important to treat the arbitrary boundary of fluid structure interaction fsi problems in computational mechanics in order to ensure complete support condition and restore the first order consistency near the boundary of smoothed particle hydrodynamics sph method for coupling finite element method fem with sph model a new ghost particle method is proposed by dividing the interceptive area of kernel support domain into subareas corresponding to boundary segments of structure the ghost particles are produced automatically for every fluid particle at each time step and the properties of ghost particles such as density mass and velocity are defined by using the subareas to satisfy the boundary condition in the coupled fem sph model the normal and shear forces from a boundary segment of structure to a fluid particle are calculated through the corresponding ghost particles and its opposite forces are exerted on the corresponding boundary segment then the momentum of the present method is conservation and there is no matching requirements between the size of elements and the size of particles the performance of the present method is discussed and validated by several fsi problems with complex geometry boundary and moving boundary

a pairwise relaxing incompressible smoothed particle hydrodynamics scheme abstract we present a formulation of incompressible smoothed particle hydrodynamics isph method that utilizes pairwise relaxing kernel to achieve approximately first order consistency previous high order formulations by using reproduced and corrected kernel function have had difficulties in ensuring momentum conservation in the new scheme relaxing constants for each kernel function are determined pair wisely throughout the entire calculation domain by enforcing the taylor series consistency condition we call this modified isph method pairwise relaxing isph or pr isph pr isph retains high order accuracy for non uniform particle distributions the spatial symmetry of the kernel function is kept in pr isph thus momentum is strictly conserved several two dimensional benchmark calculations are conducted to demonstrate the accuracy as well as the conservation property of the pr isph

variational and momentum preservation aspects of smooth particle hydrodynamic formulations this paper presents a new variational framework for various existing smooth particle hydrodynamic sph techniques and presents a new corrected sph formulation the linear and angular momentum preserving properties of sph formulations are also discussed the paper will show that in general in order to preserve angular momentum the sph equations must correctly evaluate the gradient of a linear velocity field a corrected algorithm that combines kernel correction with gradient correction is presented the paper will illustrate the theory presented with several examples relating to simple free surface flows

sph simulations of time dependent poiseuille flow at low reynolds numbers a working smoothed particle hydrodynamics sph formalism for solving the equations of motion of a viscous fluid is presented the method is based on a standard symmetrized sph expression for the viscous forces which involves only first order derivatives of the kernel through a direct evaluation of the viscous stress tensor therefore the interpolation can be performed using low order kernels of compact support without compromising the accuracy and stability of the results in principle the scheme is suitable for treating compressible fluids with arbitrary shear and bulk viscosities here we demonstrate that when it is combined with the pressure gradient correction proposed by morris et al the method is also suitable for solving the navier stokes equations for incompressible flows without any further assumptions simulations using the method show close agreement with the analytic series solutions for plane poiseuille and hagen poiseuille flows at very low reynolds numbers at least for these specific tests the results obtained are essentially independent of employing either a cubic or a quintic spline kernel

nonlinear black scholes equations in finance associated control problems and properties of solutions we study properties of solutions to fully nonlinear versions of the standard black scholes partial differential equation these equations have been introduced in financial mathematics in order to deal with illiquid markets or with stochastic volatility we show that typical nonlinear black scholes equations can be viewed as dynamic programming equation of an associated control problem we establish existence and comparison results and show that the equation induces a convex risk measure on the set of all continuous terminal value claims moreover we study the asymptotic behavior of solutions as market frictions get large finally the pricing of individual contracts relative to a book of derivatives is discussed

critical scaling for monodromy fields the large scale asymptotics of the correlations for a family of two dimensional lattice field theories is calculated at the critical temperature

lattice boltzmann model for free surface flow and its application to filling process in casting a generalized lattice boltzmann model to simulate free surface is constructed in both two and three dimensions the proposed model satisfies the interfacial boundary conditions accurately a distinctive feature of the model is that the collision processes is carried out only on the points occupied partially or fully by the fluid to maintain a sharp interfacial front the method includes an anti diffusion algorithm the unknown distribution functions at the interfacial region are constructed according to the first order chapman enskog analysis the interfacial boundary conditions are satisfied exactly by the coefficients in the chapman enskog expansion the distribution functions are naturally expressed in the local interfacial coordinates the macroscopic quantities al the interface are extracted from the least square solutions of a locally linearized system obtained from the known distribution functions the proposed method does not require any geometric front construction and is robust for any interfacial topology simulation results of realistic filling process are presented rectangular cavity in two dimensions and hammer box campbell box sheffield box and motorblock in three dimensions to enhance the stability at high reynolds numbers various upwind type schemes are developed free slip and no slip boundary conditions are also discussed

how instanton combinatorics solves painleve vi v and iiis we elaborate on a recently conjectured relation of painlev transcendents and 2d conformal field theory general solutions of painlev vi v and iii are expressed in terms of c 1 conformal blocks and their irregular limits agt related to instanton partition functions in supersymmetric gauge theories with nf 0 1 2 3 4 the resulting combinatorial series representations of painlev functions provide an efficient tool for their numerical computation at finite values of the argument the series involves sums over bipartitions which in the simplest cases coincide with gessel expansions of certain toeplitz determinants considered applications include fredholm determinants of classical integrable kernels scaled gap probability in the bulk of the gaussian unitary ensemble and all order conformal perturbation theory expansions of correlation functions in the sine gordon field theory at the free fermion point

a new open boundary formulation for incompressible sph abstract in this work a new formulation for inflow outflow boundary conditions in an incompressible smoothed particles hydrodynamics isph model is proposed it relies on the technique of unified semi analytical boundary conditions that was first proposed for wall boundary conditions in 2013 then extended to open boundaries in the framework of weakly compressible sph wcsph an isph model relying on that formulation for solid boundaries was then proposed which is the one considered here it includes a buoyancy model for temperature effects and a k turbulence closure there are two main requirements for the imposition of open boundaries in isph an algorithm to let particles enter and leave the domain and the correct imposition of open boundary conditions on the fields regarding the algorithm for particles creation destruction it relies on the variation of mass of the particles located at the open boundaries when the mass of such a particle reaches a threshold a new particle is released on the other hand the imposition of open boundary conditions on the fields is done by prescribing the value of the boundary terms appearing in the semi analytical formulation the formulation was first validated in 2 d on a cut dam break a case of propagation of a solitary wave and a creager weir it was then extended to 3 d and tested on a 3 d circular pipe a preliminary application case consisting of two connected pipes at different temperatures was then simulated the results are promising since in all cases the fluid enters and leaves the domain as prescribed and generating none or very few reflected waves

variable resolution for sph a dynamic particle coalescing and splitting scheme abstract in this paper a novel variable resolution method using particle splitting and coalescing for the sph numerical solution of the navier stokes equations is presented the key idea of the scheme is to modify dynamically the particle sizes by means of splitting and coalescing merging individual particles to provide good resolution only where it is needed the sph scheme adopted is derived using the variational principle guaranteeing that both mass and momentum are conserved for particles with different smoothing lengths a particle shifting procedure is used to prevent unacceptable anisotropic distributions of the particles and is further generalized for treating domains with variable mass particles the algorithm has been tested against analytical solutions for poiseuille and taylor green flows showing that the shifting algorithm is effective in increasing the accuracy and that error introduced by the splitting and coalescing is negligible the capability of the numerical scheme for increasing efficiency is shown for more general problems the simulations of a moving square in a box and flow past a cylinder have shown that the particle refinement procedure is able to increase the efficiency while maintaining the same level of accuracy as a uniform distribution with the most refined resolution

hamiltonian formulation of the variable h sph equations in this paper a variational formulation of smooth particle hydrodynamics for dynamic problems is presented the resulting equations treat the continuum as a hamiltonian system of particles where the constitutive equation of the material is represented via an internal energy term in the case of fluids the internal energy term is a function of density the new formulation introduces a variable smoothing length for the evaluation of density and incorporates a consistent approach for the treatment of rigid boundaries the method overcomes some problems faced by standard sph approaches that use constant smoothing lengths and provides a variational context for a variable smoothing length formulation a numerical example shows the capabilities of this novel formulation

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a self adaptive oriented particles level set method for tracking interfaces a new method for tracking evolving interfaces by lagrangian particles in conjunction with a level set approach is introduced this numerical technique is based on the use of time evolution equations for fundamental vector and tensor quantities defined on the front and represents a new and convenient way to couple the advantages of the eulerian description given by a level set function f to the use of lagrangian massless particles the term oriented points out that the information advected by the particles not only concern the spatial location but also the local outward normal vector n to the interface c and the second fundamental tensor the shape operator n the particles are exactly located upon c and provide all the requested information for tracking the interface on their own in addition a self adaptive mechanism suitably modifies at each time step the markers distribution in the numerical domain each particle behaves both as a potential seeder of new markers on c so as to guarantee an accurate reconstruction of the interface and a de seeder to avoid any useless gathering of markers and to limit the computational effort the algorithm is conceived to avoid any transport equation for f and to confine the level set function to the role of a mere post processing tool thus all the numerical diffusion problems usually affecting the level set methodology are removed the method has been tested both on 2d and 3d configurations it carries out a fast reconstruction of the interface and its accuracy is only limited by the spatial resolution of the mesh

unified semi analytical wall boundary conditions applied to 2 d incompressible sph this work aims at improving the 2 d incompressible sph model isph by adapting it to the unified semi analytical wall boundary conditions proposed by ferrand et al 10 the isph algorithm considered is as proposed by lind et al 25 based on the projection method with a divergence free velocity field and using a stabilising procedure based on particle shifting however we consider an extension of this model to reynolds averaged navier stokes equations based on the k e turbulent closure model as done in 10 the discrete sph operators are modified by the new description of the wall boundary conditions in particular a boundary term appears in the laplacian operator which makes it possible to accurately impose a von neumann pressure wall boundary condition that corresponds to impermeability the shifting and free surface detection algorithms have also been adapted to the new boundary conditions moreover a new way to compute the wall renormalisation factor in the frame of the unified semi analytical boundary conditions is proposed in order to decrease the computational time we present several verifications to the present approach including a lid driven cavity a water column collapsing on a wedge and a periodic schematic fish pass our results are compared to finite volumes methods using volume of fluids in the case of free surface flows we briefly investigate the convergence of the method and prove its ability to model complex free surface and turbulent flows the results are generally improved when compared to a weakly compressible sph model with the same boundary conditions especially in terms of pressure prediction

spin operator matrix elements in the superintegrable chiral potts quantum chain we derive spin operator matrix elements between general eigenstates of the superintegrable n symmetric chiral potts quantum chain of finite length our starting point is the extended onsager algebra recently proposed by baxter for each pair of spaces onsager sectors of the irreducible representations of the onsager algebra we calculate the spin matrix elements between the eigenstates of the hamiltonian of the quantum chain in factorized form up to an overall scalar factor this factor is known for the ground state onsager sectors for the matrix elements between the ground states of these sectors we perform the thermodynamic limit and obtain the formula for the order parameters for the ising quantum chain in a transverse field n 2 case the factorized form for the matrix elements coincides with the corresponding expressions obtained recently by the separation of variables method

simulating free surface flows with sph the sph smoothed particle hydrodynamics method is extended to deal with free surface incompressible flows the method is easy to use and examples will be given of its application to a breaking dam a bore the simulation of a wave maker and the propagation of waves towards a beach arbitrary moving boundaries can be included by modelling the boundaries by particles which repel the fluid particles the method is explicit and the time steps are therefore much shorter than required by other less flexible methods but it is robust and easy to program

dynamic refinement and boundary contact forces in sph with applications in fluid flow problems this paper presents a general method for dynamic particle refinement in smoothed particle hydrodynamics sph candidate particles are split into several daughter particles according to a given refinement pattern centred about the original particle through the solution of a non linear minimization problem the optimal mass distribution of the daughter particles is obtained so as to reduce the errors introduced to the underlying density field this procedure necessarily conserves the mass of the system conservation of energy and momentum results are also discussed copyright 2007 john wiley sons ltd

ising correlations and elliptic determinants correlation functions of the two dimensional ising model on the periodic lattice can be expressed in terms of form factors matrix elements of the spin operator in the basis of common eigenstates of the transfer matrix and translation operator free fermion structure of the model implies that any multiparticle form factor is given by the pfaffian of a matrix constructed from the two particle ones crossed two particle form factors can be obtained by inverting a block of the matrix of linear transformation induced on fermions by the spin conjugation we show that the corresponding matrix is of elliptic cauchy type and use this observation to solve the inversion problem explicitly non crossed two particle form factors are then obtained using theta functional interpolation formulas this gives a new simple proof of the factorized formulas for periodic ising form factors conjectured by a bugrij and one of the authors

a numerical method for suspension flow abstract peskin s j comput phys 25 220 1977 immersed boundary technique is modified to give a new numerical method for studying a fluid with suspended elastic particles as before the presence of the suspended particles is transmitted to the fluid through a force density term in the fluid equations as a result one set of equations holds in the entire computational domain eliminating the need to apply boundary conditions on the surface of suspended objects the new method computes the force density by discretizing the stress strain constitutive equations for an elastic solid on a grid using data provided by clusters of lagrangian points this approach clearly specifies the material properties of the suspended objects a simple data structure for the lagrangian points makes it easy to model suspended solids with arbitrary shape and size the method is validated by comparing numerical results for elastic vibrations and particle settling in viscous fluids with theory and analysis the capability of the method to do a wide range of problems is illustrated by qualitative results for lubrication and cavity flow problems

variable resolution for sph in three dimensions towards optimal splitting and coalescing for dynamic adaptivity abstract as smoothed particle hydrodynamics sph becomes increasingly popular for complex flow analysis the need to improve efficiency particularly for 3 d problems is becoming greater automatic adaptivity with variable particle size is therefore desirable in this paper a novel 3 d splitting and coalescing algorithm is developed which minimizes density error while conserving both mass and momentum using a variational principle accuracy is increased in refined areas unaffected by coarser particle distributions elsewhere for particle splitting the key criteria are the number of split daughter particles their distribution spacing and kernel size four different splitting arrangements are investigated including a cubic stencil with 8 particles a cubic stencil with an additional 6 located at the face centres an icosahedron shaped arrangement with 14 particles and a dodecahedron shaped arrangement with 20 particles where particles are located at the vertices the error analysis also examines whether retaining a particle at the centre of the arrangement is necessary revealing that regardless of the stencil adopted to minimize the density error a daughter particle should be placed at the same position of the original particle the optimum configuration is found to be the icosahedron shaped arrangement while commonly used smoothing kernels such as the cubic and quintic splines and wendland produce similar density errors so that the optimal refinement stencil is effectively independent of the kernel choice a new 3 d coalescing scheme completes the algorithm such that the particle resolution can be either increased or reduced locally the sph splitting and coalescing scheme is tested with poiseuille flow showing negligible loss of convergence accuracy in the refined area and the lid driven cavity for a wide range of reynolds number showing good agreement with reference solutions again with local accuracy defined by particle distribution

the sph equations for fluids existing smoothed particle hydrodynamics sph formulations for simulating continuous fluids have errors that may be divergent and it has been known for some time that the sph equations do not satisfy low order polynomial completeness conditions here sph equations are derived that have convergent error terms and a correction method is presented for enforcing low order polynomial completeness irrespective of how many completeness conditions are required discretization is achieved through division of the model domain in its initial state into sub domains that have lagrangian boundaries it is shown that boundary integrals appearing in one derivation of the sph equations may be treated as a convergent error in simulations of basic fluid flows convergence and zeroth order completeness are demonstrated but significant instabilities and a failure to conserve energy are observed copyright 2009 john wiley sons ltd

an implicit boundary approach for viscous compressible high reynolds flows using a hybrid remeshed particle hydrodynamics method abstract we present an implicit boundary particle method with background mesh adaptation we use a brinkman penalisation to represent the boundary of the domain and a remeshed particle method to simulate viscous flow with high reynolds numbers a penalty term is added to the navier stokes equations to impose the boundary conditions the boundary conditions are enforced to a specific precision with no need to modify the numerical method or change the grid achieving an implicit approach for flow around complex boundaries geometries simulation the main idea of the brinkman penalisation method is to model the solid obstacle as a porous medium the governing equations for the compressible fluid and penalised navier stokes for the porous medium are solved simultaneously without need for interface conditions the accuracy of the method is tested for a number of benchmark problems starting with simple cases such as a periodic laminar flow in a channel poiseuille flow to more complex problems such as the lid driven cavity with high reynolds numbers the reflection and propagation of a compressible shock wave and finally swirling flow in a model two stroke diesel engine the remeshed particle mesh method coupled with brinkman penalisation provides a good quality simulation and the results are in agreement with analytical or reference solutions

form factors in the baxter bazhanov stroganov model i norms and matrix elements we continue our investigation of the baxter bazhanov stroganov model using the method of separation of variables 1 in this paper we calculate the norms and matrix elements of a local spin operator between eigenvectors of the auxiliary problem for the norm the multiple sums over the intermediate states are performed explicitly in the case n 2 we solve the baxter equation and obtain form factors of the spin operator of the periodic ising model on a finite lattice

sph with dynamical smoothing length adjustment based on the local flow kinematics abstract due to the lagrangian nature of smoothed particle hydrodynamics sph the adaptive resolution remains a challenging task in this work we first analyse the influence of the simulation parameters and the smoothing length on solution accuracy in particular in high strain regions based on this analysis we develop a novel approach to dynamically adjust the kernel range for each sph particle separately accounting for the local flow kinematics we use the okubo weiss parameter that distinguishes the strain and vorticity dominated regions in the flow domain the proposed development is relatively simple and implies only a moderate computational overhead we validate the modified sph algorithm for a selection of two dimensional test cases the taylor green flow the vortex spin down the lid driven cavity and the dam break flow against a sharp edged obstacle the simulation results show good agreement with the reference data and improvement of the long term accuracy for unsteady flows for the lid driven cavity case the proposed dynamical adjustment remedies the problem of tensile instability particle clustering

non smooth kernels for meshfree methods in fluid dynamics a function and its first two derivatives are estimated by convolutions with well chosen non differentiable kernels the convolutions are in turn approximated by newton cotes integration techniques with the aid of a polynomial interpolation based on an arbitrary finite set of points precise numerical results are obtained with far fewer points than that in classic sph and error bounds are derived

a weighted meshfree collocation method for incompressible flows using radial basis functions abstract a weighted strong form collocation method using radial basis functions and explicit time integration is proposed to solve the incompressible navier stokes equations the velocities and pressure are solved directly at the same time step and the continuity equation is satisfied at each time step which improve the solution accuracy and stability no artificial compressibility coefficient needs to be introduced for modeling the incompressible flows and no pressure oscillation arises in the numerical solutions optimal convergence can be achieved by imposing the derived proper weights on the boundaries and the continuity equation radial basis collocation method in a lagrangian form is quite easy to capture the moving boundary or free surface in flow problems moreover solid boundary conditions can be enforced directly and no special treatments are required further critical time step for the explicit time integration is predicted in the stability analysis and the influences on the stability are evaluated numerical studies validate the high accuracy as well as good stability of the presented method

periodic ising correlations in this paper we first rework b kaufman s 1949 paper phys rev 76 1232 1949 by using representation theory our approach leads to a simpler and more direct way of deriving the spectrum of the transfer matrix for the finite periodic ising model we then determine formulas for the spin correlation functions that depend on the matrix elements of the induced rotation associated with the spin operator in a basis of eigenvectors for the transfer matrix the representation of the spin matrix elements is obtained by considering the spin operator as an intertwining map we exhibit the new elements v and v in the bugrij lisovyy formula phys lett a 319 390 2003 as part of a holomorphic factorization of the periodic and antiperiodic summability kernels on the spectral curve associated with the induced rotation for the transfer matrix

fredholm determinants jimbo miwa ueno τ functions and representation theory the authors show that a wide class of fredholm determinants arising in the representation theory of big groups such as the infinite dimensional unitary group solve painleve equations their methods are based on the theory of integrable operators and the theory of riemann hilbert problems 2002 wiley periodicals inc

sph with the multiple boundary tangent method in this article we present an improved solid boundary treatment formulation for the smoothed particle hydrodynamics sph method benchmark simulations using previously reported boundary treatments can suffer from particle penetration and may produce results that numerically blow up near solid boundaries as well current sph boundary approaches do not properly treat curved boundaries in complicated flow domains these drawbacks have been remedied in a new boundary treatment method presented in this article called the multiple boundary tangent mbt approach in this article we present two important benchmark problems to validate the developed algorithm and show that the multiple boundary tangent treatment produces results that agree with known numerical and experimental solutions the two benchmark problems chosen are the lid driven cavity problem and flow over a cylinder the sph solutions using the mbt approach and the results from literature are in very good agreement these solutions involved solid boundaries but the approach presented herein should be extendable to time evolving free surface boundaries copyright 2008 john wiley sons ltd

a high order finite difference method for option valuation in this paper we propose the use of an efficient high order finite difference algorithm to price options under several pricing models including the blackscholes model the mertons jumpdiffusion model the hestons stochastic volatility model and the nonlinear transaction costs or illiquidity models we apply a local mesh refinement strategy at the points of singularity usually found in the payoff of most financial derivatives to improve the accuracy and restore the rate of convergence of a non uniform high order five point stencil finite difference scheme for linear models the time stepping is dealt with by using an exponential time integration scheme with carathodoryfejr approximations to efficiently evaluate the product of a matrix exponential with a vector of option prices nonlinear blackscholes equations are solved using an efficient iterative scheme coupled with a richardson extrapolation our numerical experiments clearly demonstrate the high order accuracy of the proposed finite difference method making the latter a method of choice for solving both linear and nonlinear partial differential equations in financial engineering problems

control of the black scholes equation the purpose of this work is to apply the results developped by j y chemin and cl david to the black scholes equation this latter equation being directly linked to the heat equation it enables us to propose a new approach allowing to control properties of the solution by means of a shape parameter

meshless local petrov galerkin method for two dimensional nonlinear water wave problems in this paper the meshless local petrov galerkin mlpg method is extended to dealing with nonlinear water wave problems the formulation is based on general fluid governing equations and a time marching procedure at each time step the boundary value problem for the pressure is solved using the mlpg method and the velocity and position of nodes are updated by numerical integration the newly extended method is applied to simulating water waves generated by a wave maker and good agreement with analytical solutions and finite element results is presented

analysis of the incompressibility constraint in the smoothed particle hydrodynamics method summary smoothed particle hydrodynamics sph is a particle based fully lagrangian method for fluid flow simulations in this work fundamental concepts of the method are first briefly recalled then we present a thorough comparison of two different incompressibility treatments in sph the weakly compressible approach where a suitably chosen equation of state is used and the truly incompressible method in two basic variants where the velocity field projection onto a divergence free space is performed a noteworthy aspect of the study is that in each incompressibility treatment the same boundary conditions are used and further developed that allows a direct comparison to be made two dimensional and three dimensional validation cases are studied problems associated with the numerical setup are discussed and an optimal choice of the computational parameters is proposed and verified the efficiency issues of the incompressibility treatments are considered and the speed up features are highlighted the results show that the present state of the art truly incompressible methods based on a velocity correction suffer from density accumulation errors to address this issue an algorithm based on a correction for both particle velocities and positions is presented the usefulness of this density correction is examined and demonstrated copyright 2012 john wiley sons ltd

form factors of twist fields in the lattice dirac theory we study u 1 twist fields in a two dimensional lattice theory of massive dirac fermions factorized formulas for finite lattice form factors of these fields are derived using elliptic parametrization of the spectral curve of the model elliptic determinant identities and theta functional interpolation we also investigate the thermodynamic and infinite volume scaling limit where the corresponding expressions reduce to form factors of the exponential fields of the sine gordon model at the free fermion point

a variationally consistent streamline upwind petrov galerkin smooth particle hydrodynamics algorithm for large strain solid dynamics this paper presents a new smooth particle hydrodynamics sph computational framework for explicit fast solid dynamics the proposed methodology explores the use of the streamline upwind petrov galerkin supg stabilisation methodology as an alternative to the jameson schmidt turkel jst stabilisation recently presented by the authors in lee et al 2016 in the context of a conservation law formulation of fast solid dynamics the work introduced in this paper puts forward three advantageous features over the recent jst sph framework first the variationally consistent nature of the supg stabilisation allows for the introduction of a locally preserving angular momentum procedure which can be solved in a monolithic manner in conjunction with the rest of the system equations this differs from the jst sph framework where an a posteriori projection procedure was required to ensure global angular momentum preservation second evaluation of expensive harmonic and bi harmonic operators necessary for the jst stabilisation is circumvented in the new supg sph framework third the supg sph framework is more accurate for the same number of degrees of freedom than its jst sph counterpart and its accuracy is comparable to that of the robust but computationally more demanding petrov galerkin finite element method pg fem technique explored by the authors in lee et al 2014 gil et al 2014 2016 bonet et al 2015 as shown in the numerical examples included a series of numerical examples are analysed in order to benchmark and assess the robustness and effectiveness of the proposed algorithm the resulting supg sph framework is therefore accurate robust and computationally efficient three key desired features that will allow the authors in forthcoming publications to explore its applicability in large scale simulations

a consistent multi resolution smoothed particle hydrodynamics method abstract we seek to accelerate and increase the size of simulations for fluid structure interactions fsi by using multiple resolutions in the spatial discretization of the equations governing the time evolution of systems displaying two way fluid solid coupling to this end we propose a multi resolution smoothed particle hydrodynamics sph approach in which subdomains of different resolutions are directly coupled without any overlap region the second order consistent discretization of spatial differential operators is employed to ensure the accuracy of the proposed method as sph particles advect with the flow a dynamic sph particle refinement coarsening is employed via splitting merging to maintain a predefined multi resolution configuration particle regularity is enforced via a particle shifting technique to ensure accuracy and stability of the lagrangian particle based method embraced the convergence accuracy and efficiency attributes of the new method are assessed by simulating four different flows in this process the numerical results are compared to the analytical finite element and consistent sph single resolution solutions we anticipate that the proposed multi resolution method will enlarge the class of sph tractable fsi applications

modeling low reynolds number incompressible flows using sph the method of smoothed particle hydrodynamics sph is extended to model incompressible flows of low reynolds number for such flows modification of the standard sph formalism is required to minimize errors associated with the use of a quasi incompressible equation of state treatment of viscosity state equation kernel interpolation and boundary conditions are described simulations using the method show close agreement with series solutions for couette and poiseuille flows furthermore comparison with finite element solutions for flow past a regular lattice of cylinders shows close agreement for the velocity and pressure fields the sph results exhibit small pressure fluctuations near curved boundaries further improvements to the boundary conditions may be possible which will reduce these errors a similar method to that used here may permit the simulation of other flows at low reynolds numbers using sph further development will be needed for cases involving free surfaces or substantially different equations of state