Seminars

Optimal strong stability preserving time-stepping methods with upwind- and downwind-biased operators

Időpont: 
2017. 11. 30. 10:15
Hely: 
H306
Előadó: 
Yiannis Hadjimichael

A plethora of physical phenomena are modelled by hyperbolic partial differential equations, for which the exact solution is usually not known. Numerical methods are employed to approximate the solution to hyperbolic problems; however, in many cases, it is difficult to satisfy certain physical properties while maintaining high order of accuracy. Strong stability preserving (SSP) time discretizations were developed to ensure that nonlinear stability properties of the solution are maintained when coupled with suitable spatial discretizations.

In the first part of this talk, we review the development of optimal SSP Runge-Kutta and multistep methods for nonlinear problems. We emphasize the usage of an alternative representation of Runge-Kutta methods that reveals the SSP properties of such methods. Numerical examples illustrate the effectiveness and usefulness of SSP methods.
In the second part, we present some recent results related to perturbed methods that use both upwind- and downwind-biased spatial discretizations. We introduce a novel family of third-order implicit Runge–Kutta methods with arbitrarily large SSP coefficient and investigate the stability and accuracy of these methods. Moreover, we extend the analysis of SSP linear multistep methods to semi-discretized problems for which different terms on the right-hand side of the initial value problem satisfy different forward Euler (or circle) conditions. Optimal perturbed and additive monotonicity-preserving linear multistep methods are studied in the context of such problems.

Adequate numerical methods for nonlinear parabolic problems in mathematical finance

Időpont: 
2017. 11. 23. 10:15
Hely: 
H306
Előadó: 
Lubin Vulkov

The prices and hedging strategies in the real financial  market models are often described by fully nonlinear versions of the standard Black-Scholes equation. We concentrate on two classes of models: first, nonlinear Black-Scholes equations in which the volatility depends on  second space derivatives of the price(=solution) and then on regime-switching models described by systems of semilinear parabolic equations with exponential nonlinearities. The following characteristic  properties of these parabolic problems are typical: unbounded domain, boundary degeneration, maximum-minimum principle and nonnegativity preservation. We develop effective discretizations that reproduce these properties.  

What is the difference between weakly and strongly stable linear multistep methods?

Időpont: 
2017. 11. 09. 10:00
Hely: 
H306
Előadó: 
Mincsovics Miklós (BME MI, Differenciálegyenletek Tanszék)

There are two main approaches to explain the differences between them. The first one relies on the role of the parasitic roots (this is what we usually teach). The second one is more indirect and based on the general definition of stability. Spijker was the first who presented a norm pair in which the midpoint method is not stable. This example can be extended to the general weakly stable case. Finally, we upgrade this latter approach keeping its advantages and eliminating its weak point.

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