boundary regularity of rotating vortex patches we show that the boundary of a rotating vortex patch or v state in the terminology of deem and zabusky is c provided the patch is close to the bifurcation circle in the lipschitz norm the rotating patch is also convex if it is close to the bifurcation circle in the c2 norm our proof is based on burbea s approach to v states

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

fully mass conservative impes schemes for incompressible two phase flow in porous media abstract in this paper we consider fully mass conservative numerical schemes for the simulation of incompressible and immiscible two phase flow in porous media with capillary pressure compared with two kinds of conventional implicit pressure explicit saturation impes schemes the new schemes deserve a merit that the conservation of mass of both phases can be obtained the total conservation equation is obtained by the summation of the discretized conservation equation for each phase this approach is quite different from the conventional impes schemes we present two kinds of fully mass conservative impes schemes to solve the coupled systems for pressure auxiliary velocity and saturation of each phase the upwind mixed finite element methods are used to solve the pressure velocity systems which can be decoupled and the problems in the decoupled systems can be proved to be well posed moreover the new schemes are unbiased and the saturation of each phase can be proved to be bounds preserving if the time step size is smaller than a certain value the new schemes can also be applied to approximate the incompressible and immiscible two phase flow in heterogeneous porous media with different capillarity pressures several interesting examples of incompressible and immiscible two phase flow in porous media are presented to demonstrate the robustness of the new algorithms

summation by parts in time we develop a new high order accurate time integration technique for initial value problems we focus on problems that originate from a space approximation using high order finite difference methods on summation by parts form with weak boundary conditions and extend that technique to the time domain the new time integration method is global high order accurate unconditionally stable and together with the approximation in space it generates optimally sharp fully discrete energy estimates in particular it is shown how stable fully discrete high order accurate approximations of the maxwells equations the elastic wave equations and the linearized euler and navier stokes equations can obtained even though we focus on finite difference approximations we stress that the methodology is completely general and suitable for all semi discrete energy stable approximations numerical experiments show that the new technique is very accurate and has limited order reduction for stiff problems

strong stability for additive runge kutta methods space discretization of some time dependent partial differential equations gives rise to ordinary differential equations containing additive terms with different stiffness properties in these situ

imex extensions of linear multistep methods with general monotonicity and boundedness properties for solving hyperbolic systems with stiff sources or relaxation terms time stepping methods should combine favorable monotonicity properties for shocks and steep solution gradients with good stability properties for stiff terms in this paper we consider implicit explicit imex multistep methods suitable methods will be constructed based on explicit methods with general monotonicity and boundedness properties for hyperbolic equations numerical comparisons are made with several implicit explicit runge kutta methods

optimized strong stability preserving imex runge kutta methods we construct and analyze robust strong stability preserving implicit explicit runge kutta imex rk methods for models of flow with diffusion as they appear in astrophysics and in many other fields where equations with similar structure arise it turns out that besides the optimization of the region of absolute monotonicity some other properties of the methods are crucial for the success of such simulations in particular the models in our focus dictate to also take into account the step size limits associated with dissipativity positivity of the stiff parabolic terms which represent transport by diffusion the uniform convergence with respect to different stiffness properties of those same terms etc furthermore in the literature some other properties like the inclusion of a part of the imaginary axis in the stability region have been argued to be relevant in this paper we construct several new imex rk methods which differ from each other by taking various or even all of these constraints simultaneously into account it is demonstrated for some simple examples as well as for the problem of double diffusive convection that the newly constructed schemes provide a significant computational advantage over other methods from the literature due to their accumulation of different stability properties the optimized imex rk methods obtained in this paper are robust schemes that may also be useful for general models which involve the solution of advection diffusion equations or other transport equations with similar stability requirements

two classes of implicit explicit multistep methods for nonlinear stiff initial value problems the initial value problems of nonlinear ordinary differential equations which contain stiff and nonstiff terms often arise from many applications in order to reduce the computation cost implicit explicit imex methods are often applied to these problems i e the stiff and non stiff terms are discretized by using implicit and explicit methods respectively in this paper we mainly consider the nonlinear stiff initial value problems satisfying the one sided lipschitz condition and a class of singularly perturbed initial value problems and present two classes of the imex multistep methods by combining implicit one leg methods with explicit linear multistep methods and explicit one leg methods respectively the order conditions and the convergence results of these methods are obtained some efficient methods are constructed some numerical examples are given to verify the validity of the obtained theoretical results

contractivity preserving explicit linear multistep methods we investigate contractivity properties of explicit linear multistep methods in the numerical solution of ordinary differential equations the emphasis is on the general test equation frac d dt u t au t wherea is a square matrix of arbitrary orders 1 the contractivity is analysed with respect to arbitrary norms in thes dimensional space which are not necessarily generated by an inner product for given order and stepnumber we construct optimal multistep methods allowing the use of a maximal stepsize

implicit explicit one leg methods for nonlinear stiff neutral equations abstract in this paper by adapting the underlying implicit explicit imex one leg methods cf 1 2 a class of extended imex one leg eieol methods are suggested for solving nonlinear stiff neutral equations snes it is proven under some suitable conditions that eieol methods are d convergent of order 2 and stable for nonlinear snes several numerical examples are given to testify the obtained theoretical results and the computational effectiveness of eieol methods moreover a comparison with the fully implicit one leg methods is presented which shows that eieol methods have the higher computational efficiency

extrapolation based implicitexplicit peer methods with optimised stability regions in this paper we investigate a new class of implicitexplicit imex two step methods of peer type for systems of ordinary differential equations with both non stiff and stiff parts included in the source term an extrapolation approach based on already computed stage values is applied to construct imex methods with favourable stability properties optimised imex peer methods of order p 2 3 4 are given as result of a search algorithm carefully designed to balance the size of the stability regions and the extrapolation errors numerical experiments and a comparison to other implicitexplicit methods are included

high order linear multistep methods with general monotonicity and boundedness properties we consider linear multistep methods that possess general monotonicity and boundedness properties strict monotonicity in terms of arbitrary starting values for the multistep schemes is only valid for a small class of methods under very stringent step size restrictions this makes them uncompetitive with the strong stability preserving ssp runge kutta methods by relaxing these strict monotonicity requirements a larger class of methods can be considered including many methods of practical interest in this paper we construct linear multistep methods of high order up to six that possess relaxed monotonicity or boundedness properties with optimal step size conditions numerical experiments show that the new schemes perform much better than the classical monotonicity preserving multistep schemes moreover there is a substantial gain in efficiency compared to recently constructed ssp runge kutta ssprk methods

scaling to the stars a linearly scaling elliptic solver for p multigrid abstract high order methods gain increased attention in computational fluid dynamics however due to the time step restrictions arising from the semi implicit time stepping for the incompressible case the potential advantage of these methods depends critically on efficient elliptic solvers due to the operation counts of operators scaling with the polynomial degree p times the number of degrees of freedom n dof the runtime of the best available multigrid solvers scales with o p n dof this scaling with p significantly lowers the applicability of high order methods to high orders while the operators for residual evaluation can be linearized when using static condensation schwarz type smoothers require their inverses on fixed subdomains no explicit inverse is known in the condensed case and matrix matrix multiplications scale with p n dof this paper derives a matrix free explicit inverse for the static condensed operator in a cuboidal cartesian subdomain it scales with p 3 per element i e n dof globally and allows for a linearly scaling additive schwarz smoother yielding a p multigrid cycle with an operation count of o n dof the resulting solver uses fewer than four iterations for all polynomial degrees to reduce the residual by ten orders and has a runtime scaling linearly with n dof for polynomial degrees at least up to 48 furthermore the runtime is less than one microsecond per unknown over wide parameter ranges when using one core of a cpu leading to time stepping for the incompressible navier stokes equations using as much time for explicitly treated convection terms as for the elliptic solvers

asymptotic error analysis of an imex runge kutta method abstract we consider a system of singularly perturbed differential equations with singular parameter e 1 discretized with an imex runge kutta method the splitting needed for the imex method stems from a linearization of the fluxes around the limit solution we analyze the asymptotic convergence order as e 0 we show that in this setting the stage order of the implicit part of the scheme is of great importance thereby explaining earlier numerical results showing a close correlation of errors of the splitting scheme and the fully implicit one

on the asymptotic properties of imex runge kutta schemes for hyperbolic balance laws implicit explicit imex schemes are a powerful tool in the development of numerical methods for hyperbolic systems with stiff sources here we focus our attention on the asymptotic properties of such schemes like the preservation of steady states well balanced property and the behavior in presence of small space time scales asymptotic preservation property we analyze conditions under which the standard additive approach based on taking the fluxes explicitly and the sources implicitly yields a well balanced behavior in addition we consider a partitioned strategy which possesses better well balanced properties the behavior of the additive and partitioned approaches under classical scaling limits is then studied in the context of asymptotic preserving schemes additional order conditions that guarantee the correct behavior of the schemes in the navier stokes regime are derived several examples illustrate these asymptotic behaviors and the performance of the new methods

fourth order time stepping for stiff pdes on the sphere we present in this paper algorithms for solving stiff pdes on the unit sphere with spectral accuracy in space and fourth order accuracy in time these are based on a variant of the double fourier sphere method in coefficient space with multiplication matrices that differ from the usual ones and implicit explicit time stepping schemes operating in coefficient space with these new matrices allows one to use a sparse direct solver avoids the coordinate singularity and maintains smoothness at the poles while implicit explicit schemes circumvent severe restrictions on the time steps due to stiffness a comparison is made against exponential integrators and it is found that implicit explicit schemes perform best implementations in matlab and chebfun make it possible to compute the solution of many pdes to high accuracy in a very convenient fashion

third order conservative sign preserving and steady state preserving time integrations and applications in stiff multispecies and multireaction detonations abstract in this paper we develop third order conservative sign preserving and steady state preserving time integrations and seek their applications in multispecies and multireaction chemical reactive flows in this problem the density and pressure are nonnegative and the mass fraction for the ith species denoted as z i 1 i m should be between 0 and 1 where m is the total number of species there are four main difficulties in constructing high order bound preserving techniques for multispecies and multireaction detonations first of all most of the bound preserving techniques available are based on euler forward time integration therefore for problems with stiff source the time step will be significantly limited secondly the mass fraction does not satisfy a maximum principle and hence it is not easy to preserve the upper bound 1 thirdly in most of the previous works for gaseous denotation the algorithm relies on second order strang splitting methods where the flux and stiff source terms can be solved separately and the extension to high order time discretization seems to be complicated finally most of the previous ode solvers for stiff problems cannot preserve the total mass and the positivity of the numerical approximations at the same time in this paper we will construct third order conservative sign preserving rugne kutta and multistep methods to overcome all these difficulties the time integrations do not depend on the strang splitting i e we do not split the flux and the stiff source terms moreover the time discretization can handle the stiff source with large time step and preserves the steady state numerical experiments will be given to demonstrate the good performance of the bound preserving technique and the stability of the scheme for problems with stiff source terms

immersed boundary smooth extension the immersed boundary method is a simple efficient and robust numerical scheme for solving pde in general domains yet it only achieves first order spatial accuracy near embedded boundaries in this paper we introduce a new high order numerical method which we call the immersed boundary smooth extension ibse method the ibse method achieves high order accuracy by smoothly extending the unknown solution of the pde from a given smooth domain to a larger computational domain enabling the use of simple cartesian grid discretizations e g fourier spectral methods the method preserves much of the flexibility and robustness of the original ib method in particular it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta functions to communicate information between the computational grid and the boundary we present a fast algorithm for solving elliptic equations which forms the basis for simple high order implicit time methods for parabolic pde and implicit explicit methods for related nonlinear pde we apply the ibse method to solve the poisson heat burgers and fitzhugh nagumo equations and demonstrate fourth order pointwise convergence for dirichlet problems and third order pointwise convergence for neumann problems

immersed boundary smooth extension ibse a high order method for solving incompressible flows in arbitrary smooth domains abstract the immersed boundary method is a simple efficient and robust numerical scheme for solving pde in general domains yet for fluid problems it only achieves first order spatial accuracy near embedded boundaries for the velocity field and fails to converge pointwise for elements of the stress tensor in a previous work we introduced the immersed boundary smooth extension ibse method a variation of the ib method that achieves high order accuracy for elliptic pde by smoothly extending the unknown solution of the pde from a given smooth domain to a larger computational domain enabling the use of simple cartesian grid discretizations in this work we extend the ibse method to allow for the imposition of a divergence constraint and demonstrate high order convergence for the stokes and incompressible navier stokes equations up to third order pointwise convergence for the velocity field and second order pointwise convergence for all elements of the stress tensor the method is flexible to the underlying discretization we demonstrate solutions produced using both a fourier spectral discretization and a standard second order finite difference discretization

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

semi lagrangian multistep exponential integrators for index 2 differential algebraic systems implicit explicit imex multistep methods are very useful for the time discretization of convection diffusion pde problems such as the burgers equations and the incompressible navier stokes equations in the latter as well as in pde models of plasma physics and of electromechanical systems semi discretization in space gives rise to differential algebraic dae system of equations often of index higher than 1 in this paper we propose a new class of exponential integrators for index 2 daes arising from the semi discretization of pdes with a dominating and typically nonlinear convection term this class of problems includes the incompressible navier stokes equations the integration methods are based on the backward differentiation formulae bdf and they can be applied without modifications in the semi lagrangian integration of convection diffusion problems the approach gives improved performance at low viscosity regimes

implicit explicit formulations of a three dimensional nonhydrostatic unified model of the atmosphere numa we derive an implicit explicit imex formalism for the three dimensional 3d euler equations that allow a unified representation of various nonhydrostatic flow regimes including cloud resolving and mesoscale flow in a 3d cartesian domain as well as global regimes flow in spherical geometries this general imex formalism admits numerous types of methods including single stage multistep methods e g adams methods and backward difference formulas and multistage single step methods e g additive runge kutta methods the significance of this result is that it allows a numerical model to reuse the same machinery for all classes of time integration methods described in this work we also derive two classes of imex methods one dimensional and 3d and show that they achieve their expected theoretical rates of convergence regardless of the geometry e g 3d box or sphere and introduce a new second order imex runge kutta method that performs better than the other second order methods considered we

on the existence of solutions of a class of singular nonlinear two point boundary value problems we examine the existence of solutions of the class of singular nonlinear two point boundary value problems y a x y ffi f x y 0 l we show that for every a l a unique solution of the singular two point boundary value problem exists provided u kl where u sup 3f ay and k i is the first positive zero of jta l 2 fk j z is bessel s function of the first kind of order s interestingly k i is a monotonically increasing function of a values of k i for some values of a are tabulated

implicit explicit multistep methods for nonlinear parabolic equations implicit explicit multistep methods for nonlinear parabolic equations were recently analyzed in 2 3 1 in these papers the linear operator of the equation is assumed time independent self adjoint and positive definite then the linear part is discretized implicitly and the remaining part explicitly here we slightly relax the hypotheses on the linear operator by allowing part of it to be time dependent or nonself adjoint we establish optimal order a priori error estimates

linearly implicit methods for nonlinear pdes with linear dispersion and dissipation the linear stability of imex implicit explicit methods and exponential integrators for stiff systems of odes arising in the discrete solution of pdes is examined for nonlinear pdes with both linear dispersion and dissipation and a clear method of visualization of the linear stability regions is proposed predictions are made based on these visualizations and are supported by a series of experiments on five pdes including quasigeostrophic equations and stratified boussinesq equations the experiments involving 24 imex and exponential methods of third and fourth order confirm the predictions of the linear stability analysis that the methods are typically limited by small eigenvalues of the linear term and by eigenvalues on or near the imaginary axis rather than by large eigenvalues near the negative real axis the experiments also demonstrate that imex methods achieve comparable stability to exponential methods and that exponential methods are significantly more accurate only when the problem is nearly linear novel imex predictor corrector methods are also derived

stability of explicit time discretizations for solving initial value problems stability regions of explicit linear time discretization methods for solving initial value problems are treated if an integration method needsm function evaluations per time step then we scale the stability region by dividing bym we show that the scaled stability region of a method satisfying some reasonable conditions cannot be properly contained in the scaled stability region of another method bounds for the size of the stability regions for three different purposes are then given for general nonlinear ordinary differential systems for systems obtained from parabolic problems and for systems obtained from hyperbolic problems we also show how these bounds can be approached by high order methods

application of a theorem of m g krein to singular integrals on donne des estimations holderiennes et de type l 2 pour les integrales singulieres sur des espaces homogenes au sens de coifman weiss on utilise un theoreme de m g krein pour passer des estimations holderiennes aux l 2 estimations

steady state and sign preserving semi implicit runge kutta methods for odes with stiff damping term in this paper we develop a family of second order semi implicit time integration methods for systems of ordinary differential equations odes with stiff damping term the important feature of the new methods resides in the fact that they are capable of exactly preserving the steady states as well as maintaining the sign of the computed solution under the time step restriction determined by the nonstiff part of the system only the new semi implicit methods are based on the modification of explicit strong stability preserving runge kutta ssp rk methods and are proven to have a formal second order of accuracy a alpha stability and stiff decay we illustrate the performance of the proposed ssp rk based semi implicit methods on both a scalar ode example and a system of odes arising from the semi discretization of the shallow water equations with stiff friction term the obtained numerical results clearly demonstrate that the ability of the introduced ode solver to exactly preserve equilibria plays

on the v states for the generalized quasi geostrophic equations we prove the existence of the v states for the generalized inviscid sqg equations with alpha in 0 1 these structures are special rotating simply connected patches with m fold symmetry bifurcating from the trivial solution at some explicit values of the angular velocity this produces inter alia an infinite family of non stationary global solutions with uniqueness

on monotonicity and boundedness properties of linear multistep methods in this paper an analysis is provided of nonlinear monotonicity and boundedness properties for linear multistep methods instead of strict monotonicity for arbitrary starting values we shall focus on generalized monotonicity or boundedness with runge kutta starting procedures this allows many multistep methods of practical interest to be included in the theory in a related manner we also consider contractivity and stability in arbitrary norms

monotonicity preserving linear multistep methods in this paper we provide an analysis of monotonicity properties for linear multistep methods these monotonicity properties include positivity and the diminishing of total variation we also pay particular attention to related boundedness properties such as the total variation bounded tvb property in the analysis the multistep methods are considered in combination with suitable starting procedures this allows for monotonicity statements for classes of methods which are important and often used in practice but which were thus far not covered by theoretical results

error analysis of imex runge kutta methods derived from differential algebraic systems in this paper we present an error analysis of the imex runge kutta methods when applied to stiff problems containing a nonstiff term and a stiff term characterized by a small stiffness parameter varepsilon in this analysis we expand the global error in powers of varepsilon and show that the coefficients of the error are the global errors of the imex runge kutta method applied to a differential algebraic system interesting convergence results of these errors and of the remainder of the expansion allow us to determine sharp error bounds for stiff problems as a representative example of stiff problems we have chosen the van der pol equation we illustrate that the theoretical prediction is confirmed by the numerical test specifically an order reduction phenomenon is observed when the problem becomes increasingly stiff in particular making several assumptions we try to improve global error estimates of several imex runge kutta methods existing in the literature

implicit explicit multistep methods for quasilinear parabolic equations efficient combinations of implicit and explicit multistep methods for nonlinear parabolic equations were recently studied in 1 in this note we present a refined analysis to allow more general nonlinearities the abstract theory is applied to a quasilinear parabolic equation

a posteriori error analysis of imex multi step time integration methods for advection diffusion reaction equations abstract implicit explicit imex schemes are an important and widely used class of time integration methods for both parabolic and hyperbolic partial differential equations we develop accurate a posteriori error estimates for a user defined quantity of interest for two classes of multi step imex schemes for advection diffusion reaction problems the analysis proceeds by recasting the imex schemes into a variational form suitable for a posteriori error analysis employing adjoint problems and computable residuals the a posteriori estimates quantify distinct contributions from various aspects of the spatial and temporal discretizations and can be used to evaluate discretization choices numerical results are presented that demonstrate the accuracy of the estimates for a representative set of problems

higher order time stepping methods for time dependent reaction diffusion equations arising in biology abstract this paper demonstrates the use of higher order methods to solve some time dependent stiff pdes in the past the most popular numerical methods for solving system of reaction diffusion equations was based on the combination of low order finite difference method with low order time stepping method we extend in this report the compatibility of fourth order finite difference scheme in space coupled with fourth order time stepping methods such as imexlm4 imexpc4 imexrk4 and etdrk4 b in time for direct integration of reaction diffusion equations in one space dimension some interesting numerical anomaly phenomenons associated with steady state solutions of the examples chosen from the literature are well presented to address the naturally arising points and queries our findings have led to the understanding of pattern formation such as spiral waves and patchy structures as well as some spatio temporal dynamical structures

runge kutta approximation of quasi linear parabolic equations we study the convergence properties of implicit runge kutta meth ods applied to time discretization of parabolic equations with time or solution dependent operator error bounds are derived in the energy norm the con vergence analysis uses two different approaches the first technically simpler approach relies on energy estimates and requires algebraic stability of the runge kutta method the second one is based on estimates for linear time invariant equations and uses fourier and perturbation techniques it applies to a 9 stable runge kutta methods and yields the precise temporal order of conver gence this order is noninteger in general and depends on the type of boundary conditions

extrapolated implicit explicit time stepping this paper constructs extrapolated implicit explicit time stepping methods that allow one to efficiently solve problems with both stiff and nonstiff components the proposed methods are based on euler steps and can provide very high order discretizations of odes index 1 daes and pdes in the method of lines framework implicit explicit schemes based on extrapolation are simple to construct easy to implement and straightforward to parallelize this work establishes the existence of perturbed asymptotic expansions of global errors explains the convergence orders of these methods and studies their linear stability properties numerical results with stiff ode dae and pde test problems confirm the theoretical findings and illustrate the potential of these methods to solve multiphysics multiscale problems

a collocated method for the incompressible navier stokes equations inspired by the box scheme we present a new finite difference numerical method to solve the incompressible navier stokes equations using a collocated discretization in space on a logically cartesian grid the method shares some common aspects with and it was inspired by the box scheme it uses centered second order accurate finite difference approximations for the spatial derivatives combined with semi implicit time integration the proposed method is constructed to ensure discrete conservation of mass and momentum by discretizing the primitive velocity pressure form of the equations the continuity equation is enforced exactly to machine accuracy at the collocated locations whereas the momentum equations are evaluated in a staggered manner this formulation preempts the appearance of spurious pressure modes in the embedded elliptic problem associated with the pressure the method shows uniform order of accuracy both in space and time for velocity and pressure in addition the skew symmetric form of the non linear advection term of the navier stokes equations improves discrete conservation of kinetic energy in the inviscid limit to within the order of the truncation error of the time integrator the method has been formulated to accommodate different types of boundary conditions fully periodic periodic channel inflow outflow and lid driven cavity always ensuring global mass conservation a novel aspect of this finite difference formulation is the derivation of the discretization near boundaries using the weak form of the equations as in the finite element method the method of manufactured solutions is utilized to perform accuracy analysis and verification of the solver to assess the applicability of the new method presented in this paper four realistic flow problems have been simulated and results are compared with those in the literature these cases include a lid driven cavity backward facing step kovasznay flow and fully developed turbulent channel