A Semi-Lagrangian Particle Level Set Finite Element Method for Interface Problems

We present a quasi-monotone semi-Lagrangian particle level set (QMSL-PLS) method for moving interfaces. The QMSL method is a blend of first order monotone and second order semi-Lagrangian methods. The QMSL-PLS method is easy to implement, efficient, and well adapted for unstructured, either simplicial or hexahedral, meshes. We prove that it is unconditionally stable in the maximum discrete norm, � · �h,∞, and the error analysis shows that when the level set solution u(t) is in the Sobolev space Wr+1,∞(D), r ≥ 0, the convergence in the maximum norm is of the form (KT/Δt)min(1,Δt � v �h,∞ /h)((1 − α)hp + hq), p = min(2, r + 1), and q = min(3, r + 1),where v is a velocity. This means that at high CFL numbers, that is, when Δt > h, the error is O( (1−α)hp+hq) Δt ), whereas at CFL numbers less than 1, the error is O((1 − α)hp−1 + hq−1)). We have tested our method with satisfactory results in benchmark problems such as the Zalesak’s slotted disk, the single vortex flow, and the rising bubble

A finite element semi-Lagrangian explicit Runge–Kutta–Chebyshev method for convection dominated reaction–diffusion problems

AbstractExplicit Runge–Kutta–Chebyshev methods have proved to be efficient for reaction–diffusion problems of moderate stiffness. In this paper, we extend such an efficiency to convection-dominated-reaction–diffusion problems by giving a formulation of these methods in a semi-Lagrangian framework, using C0-finite elements of degree m⩾2 as the space discretization method. We also study the convergence in the L2-norm of the methods proposed in this paper

Multiscale modeling of viscoelastic fluids: an up-to-date CONNFFESSIT

The present communication introduces an up-to-date version of the CONNFFESSIT method in the field of micro-macro simulations of non-Newtonian fluids. The ‘macro’ section employs a semi-Lagrangian method in order to reduce the Navier-Stokes equations to a Stokes-like subproblem. Linear systems arising from the finite element formulation are solved via the ‘Incomplete Cholesky Conjugate Gradient’ iterative algorithm, wherein the sparsity pattern of the matrices is taking into account. As to the ‘micro’ part, the stochastic formulation simplifies the Fokker-Planck equations in the configuration space to stochastic differential equations for the internal degrees of freedom of the particles (‘dumbbells’) conveying the rheological information of the kinetic model, their integration being accomplished by means of a semi-implicit, PredictorCorrector algorithm. The ‘micro-macro’ coupling involves the polymer stress tensor, which is computed through a mixed ‘Finite Element / Natural Element’ method.An extended, search-and-locate method for unstructured meshes and non-connected domains has been implemented. The robustness and efficiency of the method is highlighted on a benchmark problem (10:1 planar contraction)

A second order in time modified Lagrange-Galerkin finite element method for the incompressible Navier-Stokes equations

We introduce a second order in time modified Lagrange--Galerkin (MLG) method for the time dependent incompressible Navier--Stokes equations. The main ingredient of the new method is the scheme proposed to calculate in a more efficient manner the Galerkin projection of the functions transported along the characteristic curves of the transport operator. We present error estimates for velocity and pressure in the framework of mixed finite elements when either the mini-element or the $P2/P1$ Taylor--Hood element are used

Stochastic semi-Lagrangian micro–macro calculations of liquid crystalline solutions in complex flows

A general method for the simulation of complex flows of liquid crystalline polymers (LCPs) using a stochastic semi-Lagrangian micro–macro method is introduced. The macroscopic part uses a spatial-temporal second order accurate semi-Lagrangian algorithm, where ideas from the finite element and natural element methods are mixed in order to compute average quantities. The microscopic part employs a stochastic interpretation of the Doi–Hess LCP model, which is discretized with a second order Richardson extrapolated Euler–Maruyama scheme. The new method is validated and tested using the benchmark problem of flow between rotating eccentric cylinders. In a decoupled analysis, a discussion on the sensibility of the scalar order parameter to the macroscopic flow is offered. For the coupled situation, the proposed method predicts disclinations at certain regions of the geometry, as well as an accentuated abatement of the flow as the strength of the micro–macro interaction increases. Further examples are provided at different Peclet and concentration numbers to gain insight on the behavior of complex flows of LCPs in the eccentric cylinder geometry. The generality and robustness of the method, as well as its accurate prediction of LCP behavior under complex flows are main features of the implementatio

A subgrid viscosity Lagrance-Galerkin method for convection-diffusion problems

We present and analyze a subgrid viscosity Lagrange-Galerk in method that combines the subgrid eddy viscosity method proposed in W. Layton, A connection between subgrid scale eddy viscosity and mixed methods. Appl. Math. Comp., 133: 14 7-157, 2002, and a conventional Lagrange-Galerkin method in the framework of P1⊕ cubic bubble finite elements. This results in an efficient and easy to implement stabilized method for convection dominated convection diffusion reaction problems. Numerical experiments support the numerical analysis results and show that the new method is more accurate than the conventional Lagrange-Galerkin one

How does easing liquidity constraints affect aggregate employment?

We measure the impact of removing liquidity constraints on aggregate employment by focusing on a sudden and unexpected large liquidity injection to Spanish firms in early 2012, when the Spanish central government paid all invoices of firms to regional and municipal governments that were in arrears. We identify the effect on employment from the cross-sectional variation in the size of the liquidity injection received by Spanish municipalities. Our preliminary finding sindicate that labor market responses can be detected both in the municipality where the liquidity injection occurs and in the municipality where firms are headquartered. We find evidence that the effect on unemployment is stronger where the liquidity injection originates whereas the effect on employment is stronger where the firms are located

Un esquema adaptativo semilagrangiano con elementos finitos anisótropos para la resolución de problemas de combustión.

En este trabajo se presenta un método numérico adaptativo para la resolución de problemas temporales de Combustión en la Mecánica de Fluidos. La dis- cretización espacial de las ecuaciones está basada en el método de los elementos finitos y la discretización temporal se realiza con un esquema semilagrangiano para tratar de forma eficiente los términos convectivos. La caracteística principal de la adaptación local, es que el mallado que se construye es anisótropo, lo que le capacita para adaptarse mejor a capas límitee con un reducido coste computacional. Para ello, se define un tensor métrico en cada paso de tiempo, basado en indicadores de error construidos a priori y a posteriori. Ilustraremos el buen comportamiento del código numérico con la modelización de un problema de combustión en 2D y 3D, donde se analilzará la interacción de llamas de difusión de Hidrógeno y vórtices que pueden ser generados en un flujo turbulento

A semi-Lagrangian micro-macro method for viscoelastic flow calculations

We present in this paper a semi-Lagrangian algorithm to calculate the viscoelastic flow in which a dilute polymer solution is modeled by the FENE dumbbell kinetic model. In this algorithm the material derivative operator of the Navier–Stokes equations (the macroscopic flow equations) is discretized in time by a semi-Lagrangian formulation of the second order backward difference formula (BDF2). This discretization leads to solving each time step a linear generalized Stokes problem. For the stochastic differential equations of the microscopic scale model, we use the second order predictor-corrector scheme proposed in [22] applied along the forward trajectories of the center of mass of the dumbbells. Important features of the algorithm are (1) the new semi-Lagrangian projection scheme; (2) the scheme to move and locate both the mesh-points and the dumbbells; and (3) the calculation and space discretization of the polymer stress. The algorithm has been tested on the 2d 10:1 contraction benchmark problem and has proved to be accurate and stable, being able to deal with flows at high Weissenberg (Wi) numbers; specifically, by adjusting the size of the time step we obtain solutions at Wi=444