We study a novel class of numerical integrators, the adapted nested force-gradient schemes, used within the molecular dynamics step of the Hybrid Monte Carlo (HMC) algorithm. We test these methods in the Schwinger model on the lattice, a well known benchmark problem. We derive the analytical basis of nested force-gradient type methods and demonstrate the advantage of the proposed approach, namely reduced computational costs compared with other numerical integration schemes in HMC.

We present a new splitting method for time-dependent convention-dominated diffusion problems. The original convention diffusion system is split into two sub-systems: a pure convection system and a diffusion system. At each time step, a convection problem and a diffusion problem are solved successively. A few important features of the scheme lie in the facts that the convection subproblem is solved explicitly and multistep techniques can be used to essentially enlarge the stability region so that the resulting scheme behaves like an unconditionally stable scheme; while the diffusion subproblem is always self-adjoint and coercive so that they can be solved efficiently using many existing optimal preconditioned iterative solvers. The scheme can be extended for solving the Navier-Stokes equations, where the nonlinearity is resolved by a linear explicit multistep scheme at the convection step, while only a generalized Stokes problem is needed to solve at the diffusion step and the major stiffness matrix stays invariant in the time marching process. Numerical simulations are presented to demonstrate the stability, convergence and performance of the single-step and multistep variants of the new scheme.

The aim of this work is to compare a new uncoupled solver for the cardiac Bidomain model with a usual coupled solver. The Bidomain model describes the bioelectric activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction-diffusion partial differential equation (PDE) and an elliptic linear PDE. This system models at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue. The evolution equation is coupled through the non-linear reaction term with a stiff system of ordinary differential equations (ODEs), the so-called membrane model, describing the ionic currents through the cellular membrane. A novel uncoupled solver for the Bidomain system is here introduced, based on solving twice the parabolic PDE and once the elliptic PDE at each time step, and it is compared with a usual coupled solver. Three-dimensional numerical tests have been performed in order to show that the proposed uncoupled method has the same accuracy of the coupled strategy. Parallel numerical tests on structured meshes have also shown that the uncoupled technique is as scalable as the coupled one. Moreover, the conjugate gradient method preconditioned by Multilevel Hybrid Schwarz preconditioners converges faster for the linear systems deriving from the uncoupled method than from the coupled one. Finally, in all parallel numerical tests considered, the uncoupled technique proposed is always about two or three times faster than the coupled approach.

In this paper, we investigate the coupling between operator splitting techniques and a timeparallelization scheme, the parareal algorithm,as a numericalstrategy for the simulation of reaction-diffusion equations modelling multi-scale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reactive fronts,spatially very localized.In a series of previous studies, the numerical analysis of theoperator splitting as well as the parareal algorithmhas been conducted and such approaches have shown a great potential in the framework of reaction-diffusion and convection-diffusion-reaction systems. However, complementary studies are needed for a more complete characterizationof such techniques for these stiff configurations.Therefore,we conduct in this work a precise numerical analysis that considers thecombination of time operator splitting and the parareal algorithmin the context of stiff reaction fronts. The impact of the stiffness featured by these frontson the convergence of the method is thus quantified, and allows to conclude on an optimal strategy for the resolution ofsuch problems.We finally perform some numerical simulations in the field of nonlinear chemical dynamics that validate the theoretical estimatesand examine the performance of such strategiesin the context of academical one-dimensional test casesas well as multi-dimensional configurationssimulated on parallel architecture.

We present an iterative method based on an infinite dimensional adaptation of thesuccessive overrelaxation (SOR) algorithm for solving the 2-D neutron transport equation.In a wide range of application, the neutron transport operator admits a Self-Adjoint andm-Accretive Splitting (SAS). This splitting leads to an ADI-like iterative method whichconverges unconditionally and is equivalent to a fixed point problem where the operator isa 2 by 2 matrix of operators. An infinite dimensional adaptation of a SOR algorithm isthen applied to solve the matrix operator equation. Theoretical and numerical results ofconvergence are given

Skeletal patterning in the vertebrate limb,i.e., the spatiotemporal regulation of cartilage differentiation(chondrogenesis) during embryogenesis and regeneration, is oneof the best studied examples of a multicellular developmental process.Recently [Alber et al., The morphostatic limit for a model ofskeletal pattern formation in the vertebrate limb, Bulletin ofMathematical Biology, 2008, v70, pp. 460-483], a simplified two-equationreaction-diffusion system was developed to describe the interaction of two ofthe key morphogens: the activator and an activator-dependent inhibitor ofprecartilage condensation formation. A discontinuous Galerkin (DG)finite element method was applied to solve this nonlinear system on complexdomains to study the effects of domain geometry on the pattern generated [Zhu etal., Application of Discontinuous Galerkin Methods for reaction-diffusionsystems in developmental biology, Journal of Scientific Computing, 2009, v40,pp. 391-418]. In this paper, we extend these previous results and develop a DGfinite element model in a moving and deforming domain for skeletal patternformation in the vertebrate limb. Simulations reflect the actual dynamics oflimb development and indicate the important role played by the geometryof the undifferentiated apical zone.

In this paper we solve the time-dependent incompressible Navier-Stokesequations by splitting the non-linearity and incompressibility, andusing discontinuous or continuous finite element methods in space. Weprove optimal error estimates for the velocity and suboptimalestimates for the pressure. We present some numerical experiments.