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This paper is concerned with the numerical approximations of Cauchy problems for one-dimensional nonconservative hyperbolic systems. The first goal is to introduce a general concept of well-balancing for numerical schemes solving this kind of systems. Once this concept stated, we investigate the well-balance properties of numerical schemes based on the generalized Roe linearizations introduced by [Toumi, J. Comp. Phys. 102 (1992) 360–373]. Next, this general theory is applied to obtain well-balanced...
This paper is concerned with the numerical approximations of Cauchy problems for
one-dimensional nonconservative hyperbolic systems.
The first goal is to introduce a general concept of well-balancing
for numerical schemes solving this kind of systems. Once this concept stated, we
investigate the well-balance properties of numerical schemes based on the
generalized Roe linearizations introduced by [Toumi, J. Comp. Phys.102 (1992) 360–373]. Next, this general theory
is applied to obtain well-balanced...
We propose a theoretical framework for solving a class of worst scenario problems. The existence of the worst scenario is proved through the convergence of a sequence of approximate worst scenarios. The main convergence theorem modifies and corrects the relevant results already published in literature. The theoretical framework is applied to a particular problem with an uncertain boundary value problem for a nonlinear ordinary differential equation with an uncertain coefficient.
We apply a theoretical framework for solving a class of worst scenario problems to a problem with a nonlinear partial differential equation. In contrast to the one-dimensional problem investigated by P. Harasim in Appl. Math. 53 (2008), No. 6, 583–598, the two-dimensional problem requires stronger assumptions restricting the admissible set to ensure the monotonicity of the nonlinear operator in the examined state problem, and, as a result, to show the existence and uniqueness of the state solution....
As a numerical method for solving ordinary differential equations , the improved Euler method is not assumed to give exact solutions. In this paper we classify all cases where this method gives the exact solution for all initial conditions. We reduce an infinite system of partial differential equations for to a finite system that is sufficient and necessary for the improved Euler method to give the exact solution. The improved Euler method is the simplest explicit second order Runge-Kutta method....
The solvability of time-harmonic Maxwell equations in the 3D-case with nonhomogeneous conductivities is considered by adapting Sobolev space technique and variational formulation of the problem in question. Moreover, a finite element approximation is presented in the 3D-case together with an error estimate in the energy norm. Some remarks are given to the 2D-problem arising from geophysics.
In this paper the author establishes estimation of the total truncation error after steps in the fifth order Ruge-Kutta-Huťa formula for systems of differential equations. The approach is analogous to that used by Vejvoda for the estimation of the classical formulas of the Runge-Kutta type of the 4-th order.
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