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Highly anisotropic nonlinear temperature balance equation and its numerical solution using asymptotic-preserving schemes of second order in time

Alexei Lozinski, Jacek Narski, Claudia Negulescu (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This paper deals with the numerical study of a nonlinear, strongly anisotropic heat equation. The use of standard schemes in this situation leads to poor results, due to the high anisotropy. An Asymptotic-Preserving method is introduced in this paper, which is second-order accurate in both, temporal and spacial variables. The discretization in time is done using an L-stable Runge−Kutta scheme. The convergence of the method is shown to be independent of the anisotropy parameter , and this for fixed...

Identification of parameters in initial value problems for ordinary differential equations

Chleboun, Jan, Mikeš, Karel (2015)

Programs and Algorithms of Numerical Mathematics

Scalar parameter values as well as initial condition values are to be identified in initial value problems for ordinary differential equations (ODE). To achieve this goal, computer algebra tools are combined with numerical tools in the MATLAB environment. The best fit is obtained through the minimization of the summed squares of the difference between measured data and ODE solution. The minimization is based on a gradient algorithm where the gradient of the summed squares is calculated either numerically...

Identification problem for nonlinear beam -- extension for different types of boundary conditions

Radová, Jana, Machalová, Jitka (2023)

Programs and Algorithms of Numerical Mathematics

Identification problem is a framework of mathematical problems dealing with the search for optimal values of the unknown coefficients of the considered model. Using experimentally measured data, the aim of this work is to determine the coefficients of the given differential equation. This paper deals with the extension of the continuous dependence results for the Gao beam identification problem with different types of boundary conditions by using appropriate analytical inequalities with a special...

Implicit Runge-Kutta methods for transferable differential-algebraic equations

M. Arnold (1994)

Banach Center Publications

The numerical solution of transferable differential-algebraic equations (DAE’s) by implicit Runge-Kutta methods (IRK) is studied. If the matrix of coefficients of an IRK is non-singular then the arising systems of nonlinear equations are uniquely solvable. These methods are proved to be stable if an additional contractivity condition is satisfied. For transferable DAE’s with smooth solution we get convergence of order m i n ( k E , k I + 1 ) , where k E is the classical order of the IRK and k I is the stage order. For transferable...

Implicit-explicit Runge–Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations

Erik Burman, Alexandre Ern (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We analyze a two-stage implicit-explicit Runge–Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on L2-energy estimates on discrete functions in physical space. Our main...

Implicit-explicit Runge–Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations

Erik Burman, Alexandre Ern (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We analyze a two-stage implicit-explicit Runge–Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on L2-energy estimates on discrete functions in physical space. Our main...

Implicit-explicit Runge–Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations

Erik Burman, Alexandre Ern (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We analyze a two-stage implicit-explicit Runge–Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on L2-energy estimates on discrete functions in physical space. Our main...

Including eigenvalues of the plane Orr-Sommerfeld problem

Peter P. Klein (1993)

Applications of Mathematics

In an earlier paper [5] a method for eigenvalue inclussion using a Gerschgorin type theory originating from Donnelly [2] was applied to the plane Orr-Sommerfeld problem in the case of a pure Poiseuile flow. In this paper the same method will be used to deal Poiseuile and Couette flow. Potter [6] has treated this case before with an approximative method.

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