Václav Šátek

Efficient methods for solving the Stokes problem with slip boundary conditions

The paper deals with the Stokes flow with the threshold slip boundary conditions. A finite element approximation of the problem leads to the minimization of a non-differentiable energy functional subject to two linear equality constraints: the impermeability condition on the slip part of the boundary and the incompressibility of the fluid. Eliminating the velocity components, one gets the smooth dual functional in terms of three Lagrange multipliers. The first Lagrange multiplier regularizes the problem. Its components are subject to simple bounds. The other two Lagrange multipliers treat the impermeability and the incompressibility conditions. The last Lagrange multiplier represents the pressure in the whole domain. The solution to the dual problem is computed by an active set strategy and a path-following variant of the interior-point method. Numerical experiments illustrate computational efficiency.

New Trends in Taylor Series Based Computations

The paper is a part of student cooperation in AKTION project (Austria-Czech) and concentrates on the numerical solution of partial differential equations (PDEs) using high-order forward, backward and symmetrical formulas. As an example, the hyperbolic PDE is analyzed. The paper is based on the numerical solution of ordinary differential equations by the Taylor series method and on the simulation language TKSL that has been created to test the properties of the technical initial problems and to test an algorithm for Taylor series method. The idea of parallel computations using special integrators is also a part of the paper.

Application of the Modern Taylor Series Method to a multi-torsion chain

Abstract In this paper the application of a novel high accuracy numerical integration method is presented for a practical mechanical engineering application. It is based on the direct use of the Taylor series. The main idea is a dynamic automatic order setting, i.e. using as many Taylor series terms for computing as needed to achieve the required accuracy. Previous results have already proved that this numerical solver is both very accurate and fast. In this paper the performance is validated for a real engineering assembly and compared to a Jacobian power series method. The chosen experiment setup is a multi-torsional oscillator chain which reproduces typical dynamic behavior of industrial mechanical engineering problems. Its rotatory dynamics are described by linear differential equations. For the test series the system is operated in a closed-loop configuration. A reference solution of the linear differential equations of the closed-loop system for the output variable is obtained with the mathematical software tool Maple and validated by comparison to measurements from the experiment. The performance of the Modern Taylor Series Method is demonstrated by comparison to standard fixed-step numerical integration methods from the software tool Matlab/Simulink and to the Jacobian power series approximation. Furthermore, the improvement in numerical accuracy as well as stability is illustrated and CPU-times for the different methods are given.

Parallel computations based on automatic transformation of ordinary differential equations

The paper is a part of student cooperation in AKTION project (Austria-Czech). Theoretical work on the numerical solution of ordinary differential equations by the Taylor series method has been going on for a number of years. The simulation language TKSL has been created to test the properties of the technical initial problems and to test an algorithm for Taylor series method [1]. The Residue Number System (RNS) has great potential for accelerating arithmetic operation, achieved by breaking operands into several residues and computing with them independently.

Taylor series based computations and MATLAB ODE solvers comparisons

The paper is a part of student cooperation in AKTION project (Austria-Czech). Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. The paper deals with possibilities of numerical solution of Initial Value Problems of Ordinary Differential Equations (ODEs) - using the Taylor series method with automatic computation of higher Taylor series terms. The explicit and implicit scheme of Taylor series method is compared with numerical solvers implemented in MATLAB software [1]. The computation time and accuracy of our approach are compared with that of MATLAB ode solvers on a set of ODEs test examples [2].

Explicit and Implicit Taylor Series Based Computations

Motto: There are at least two ways to combat stiffness. One is to design a better computer, the other, to design a better algorithm.

25th Anniversary of TKSL

In recent years, intensive research has been done at the Brno University of Technology Faculty of Information Technology Department of Intelligent Systems in the field of numerical solutions of systems of ordinary and partial differential equations. The basic numerical method employed is the so‐called Modern Taylor Series Method (MTSM). It has been described, studied, and numerous aspects have been investigated such as processing in parallel systems. Also a simulation system TKSL has been developed which is based on the Taylor series method. For some results see [1], [2]. Although there have been considerable practical results, theoretical issues are yet to be investigated since the underlying method has been devised by analogy with analogue solvers of such systems. In this paper we provide only a basic idea of a theoretical background.The MTSM is based on a transformation of the initial problem into another initial problem with polynomials on the right‐hand sides. This is a precondition for a Taylor seri...

Stokes system with solution-dependent threshold slip boundary conditions: Analysis, approximation and implementation

The paper analyzes the Stokes system with threshold slip boundary conditions of Navier type. Based on the fixed-point formulation we prove the existence of a solution for a class of solution-dependent slip functions g satisfying an appropriate growth condition and its uniqueness provided that g is one-sided Lipschitz continuous. Further we study under which conditions the respective fixed-point mapping is contractive. To discretize the problem we use P1-bubble/P1 elements. Properties of discrete models in dependence on the discretization parameter are analysed and convergence results are established. In the second part of the paper we briefly describe the duality approach used in computations and present results of a model example.

Adaptive solution of Laplace equation

The paper is a part of student cooperation in AKTION project (Austria-Czech). Method of aposteriori error estimation based on weighted averaging to improve initial triangulation to get better solution of the planar elliptic boundary-value problem of second order and numerical illustrations of the method are presented in the paper.

Parallel computations based on modified numerical integration methods

Even though the idea of parallel computing and parallel connection of high amount of microprocessors is attractive, it is not easy to reach big increase in performance compared to single processor approach. The potential of parallel data processing has already been studied. It was found, that even a small percentage of sequential steps may lead to high reduction of performance of the entire system. This is the consequence of the fact, that most algorithms were not developed for heavy parallel systems.