• Abstract

We give an energy stability analysis of a first order, 2 step partitioned time discretization of systems of evolution equations. The method requires only uncoupled solutions of sub-systems at every time step without iteration, is long time stable and applies to general system couplings. We give a proof of long time energy stability under a time step restriction relating the time step to the size of the coupling terms.

• Abstract

In this paper we study on a Caccioppoli inequality of the unsteady Stokes system.

• Abstract

We propose an analytic approximation formula for pricing zero-coupon bonds in the case when the short-term interest rate is driven by a one-factor mean-reverting process with a volatility proportional to the power the interest rate itself. We derive its order of accuracy. Afterwards, we suggest its use in calibration and show that it can be reduced to a simple optimization problem. To test the calibration methodology, we use the simulated data from the Cox-Ingersoll- Ross model where the exact bond prices can be computed. We show that using the approximation in the calibration recovers the parameters with a high precision.

• Abstract

This paper is devoted to the study of finite element approximations of variational inequalities with a special nonlinearity coming from boundary conditions. After re-writing the problems in the form of variational inequalities, a fixed point strategy is used to show existence of solutions. Next we prove that the finite element approximations for the Stokes and Navier Stokes equations converge respectively to the solutions of each continuous problems. Finally, Uzawa's algorithm is formulated and convergence of the procedure is shown, and numerical validation test is achieved.

• Abstract

A non-overlapping domain decomposition (DD) method is used to solve a heterogeneous flow model which combines viscous flow and potential flow. Finite element method (FEM) and boundary element method (BEM) approximate the solutions to Navier-Stokes equations in the viscous flow subdomain and to Laplace equation in the potential flow subdomain, respectively. At the interface, the matching conditions involve pressure and velocity, and Bernoulli's equation gives an ordinary differential equation (ODE) defined on the interface. Algebraic formulations of the iterative schemes to solve the coupled problem are developed, and both explicit and implicit schemes can be constructed following the strategy of the Dirichlet-Neumann (D-N) method. Numerical examples using the explicit scheme implementation are reported and compared against previous experimental and/or numerical results.

• Abstract

The dam break problem shallow approximation for laminar flows of power-law non-Newtonian fluids is numerically revisited under a time and space second order adaptive method. Theoretical solutions are compared with experimental measurements from the literature and new ones made of silicon. Asymptotic behaviours are solved numerically and from autosimilar solutions. The obtained theoretical results are finally compared with experiments. These comparisons confirm the validity of the shallow approximation equations for non-Newtonian fluids subject to the horizontal dam break problem.

• Abstract

This paper is an experimental continuation of [7], where we presented one realization of a locally-conservative Eulerian-Lagrangian finite element method (LCELM) for a semilinear parabolic equation and proved an optimal convergence rate. In this paper, we present two higher- order extensions of the method of [7], along with one lower-order procedure. We show some numerical results to illustrate the accuracy and efficiency of the LCELM procedures. Optimal convergence rates for each method will be presented.