scholarly journals On the backward Euler method for a generalized Ait-Sahalia-type rate model with Poisson jumps

2020 ◽  
Author(s):  
Yuying Zhao ◽  
Xiaojie Wang ◽  
Mengchao Wang
Keyword(s):  
Euler Method ◽  
Backward Euler Method ◽  
Rate Model ◽  
Poisson Jumps ◽  
Backward Euler ◽  
Type Rate

Split-Step Backward Euler Method for Stochastic Delay Hopfield Neural Networks with Markovian Switching

2011 ◽  
Vol 58-60 ◽  
pp. 1390-1395
Author(s):  
Rong Hua Li ◽  
Li Yang ◽  
Jia Wei Li
Keyword(s):  
Neural Networks ◽  
Euler Method ◽  
Hopfield Neural Networks ◽  
Markovian Switching ◽  
Backward Euler Method ◽  
Approximation Solution ◽  
True Solution ◽  
Stochastic Delay ◽  
Backward Euler ◽  
Exponentially Stable

In this paper, split-step backward Euler method for stochastic delay Hopfield neural networks with Markovian switching is considered. The main aim of this paper is to show that the numerical approximation solution is convergent to the true solution with order. The conditions under which the numerical solution is exponentially stable in mean square are given. An example is provided for illustration.

Stable Non-symmetric Coupling of the Finite Volume Method and the Boundary Element Method for Convection-Dominated Parabolic-Elliptic Interface Problems

2020 ◽  
Vol 20 (2) ◽  
pp. 251-272
Author(s):  
Christoph Erath ◽  
Robert Schorr
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Time Discretization ◽  
Euler Method ◽  
Interface Problems ◽  
Backward Euler Method ◽  
Elliptic Interface Problems ◽  
Practical Applications ◽  
Backward Euler ◽  
Volume Method

AbstractMany problems in electrical engineering or fluid mechanics can be modeled by parabolic-elliptic interface problems, where the domain for the exterior elliptic problem might be unbounded. A possibility to solve this class of problems numerically is the non-symmetric coupling of finite elements (FEM) and boundary elements (BEM) analyzed in [H. Egger, C. Erath and R. Schorr, On the nonsymmetric coupling method for parabolic-elliptic interface problems, SIAM J. Numer. Anal. 56 2018, 6, 3510–3533]. If, for example, the interior problem represents a fluid, this method is not appropriate since FEM in general lacks conservation of numerical fluxes and in case of convection dominance also stability. A possible remedy to guarantee both is the use of the vertex-centered finite volume method (FVM) with an upwind stabilization option. Thus, we propose a (non-symmetric) coupling of FVM and BEM for a semi-discretization of the underlying problem. For the subsequent time discretization we introduce two options: a variant of the backward Euler method which allows us to develop an analysis under minimal regularity assumptions and the classical backward Euler method. We analyze both, the semi-discrete and the fully-discrete system, in terms of convergence and error estimates. Some numerical examples illustrate the theoretical findings and give some ideas for practical applications.

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