Gerhard Wanner (born 1942 in Innsbruck) [1] is an Austrian mathematician.
Gerhard Wanner (born 1942 in Innsbruck) [1] is an Austrian mathematician.
Wanner grew up in Seefeld in Tirol and studied mathematics at the University of Innsbruck, where he received his doctorate in 1965 with advisor Wolfgang Gröbner and dissertation Ein Beitrag zur numerischen Behandlung von Randwertproblemen gewöhnlicher Differentialgleichungen (A contribution to the numerical treatment of boundary value problems of ordinary differential equations). [2] He taught in Innsbruck and from 1973 at the University of Geneva.
Wanner's research deals with numerical analysis of ordinary differential equations (about which he wrote a two-volume monograph with Ernst Hairer). Wanner is the co-author of an analysis undergraduate textbook and a geometry undergraduate textbook, both of which give historically oriented explanations of mathematics.
In 2003 he was awarded, jointly with Ernst Hairer, the Peter Henrici Prize. In 2015 Wanner received SIAM's George Pólya Prize for Mathematical Exposition. [3]
He was president of the Swiss Mathematical Society from 1998 to 1999.
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In numerical analysis, the Runge–Kutta methods are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta.
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals.
Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The process continues with subsequent steps to map out the solution. Single-step methods refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt to gain efficiency by keeping and using the information from previous steps rather than discarding it. Consequently, multistep methods refer to several previous points and derivative values. In the case of linear multistep methods, a linear combination of the previous points and derivative values is used.
In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution.
In the mathematical field of numerical ordinary differential equations, a geometric integrator is a numerical method that preserves geometric properties of the exact flow of a differential equation.
In mathematics and computational science, the Euler method is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis.
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite-dimensional space of candidate solutions and a number of points in the domain, and to select that solution which satisfies the given equation at the collocation points.
In mathematics, the Runge–Kutta–Fehlberg method is an algorithm in numerical analysis for the numerical solution of ordinary differential equations. It was developed by the German mathematician Erwin Fehlberg and is based on the large class of Runge–Kutta methods.
In numerical analysis, the Dormand–Prince (RKDP) method or DOPRI method, is an embedded method for solving ordinary differential equations. The method is a member of the Runge–Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth-order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration methods are Fehlberg (RKF) and Cash–Karp (RKCK).
In numerical analysis, the Bulirsch–Stoer algorithm is a method for the numerical solution of ordinary differential equations which combines three powerful ideas: Richardson extrapolation, the use of rational function extrapolation in Richardson-type applications, and the modified midpoint method, to obtain numerical solutions to ordinary differential equations (ODEs) with high accuracy and comparatively little computational effort. It is named after Roland Bulirsch and Josef Stoer. It is sometimes called the Gragg–Bulirsch–Stoer (GBS) algorithm because of the importance of a result about the error function of the modified midpoint method, due to William B. Gragg.
In the theory of partial differential equations, an a priori estimate is an estimate for the size of a solution or its derivatives of a partial differential equation. A priori is Latin for "from before" and refers to the fact that the estimate for the solution is derived before the solution is known to exist. One reason for their importance is that if one can prove an a priori estimate for solutions of a differential equation, then it is often possible to prove that solutions exist using the continuity method or a fixed point theorem.
Ernst Hairer is a professor of mathematics at the University of Geneva known for his work in numerical analysis.
Within mathematics regarding differential equations, L-stability is a special case of A-stability, a property of Runge–Kutta methods for solving ordinary differential equations. A method is L-stable if it is A-stable and as , where is the stability function of the method. L-stable methods are in general very good at integrating stiff equations.
The following is a timeline of scientific computing, also known as computational science.
Exponential integrators are a class of numerical methods for the solution of ordinary differential equations, specifically initial value problems. This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. Because the linear part is integrated exactly, this can help to mitigate the stiffness of a differential equation. Exponential integrators can be constructed to be explicit or implicit for numerical ordinary differential equations or serve as the time integrator for numerical partial differential equations.
Marlis Hochbruck is a German applied mathematician and numerical analyst known for her research on matrix exponentials, exponential integrators, and their applications to the numerical solution of differential equations. She is a professor in the Institute for Applied and Numerical Mathematics at the Karlsruhe Institute of Technology.
Christian Lubich is an Austrian mathematician, specializing in numerical analysis.
Assyr Abdulle was a Swiss mathematician. He specialized in numerical mathematics.
Zero-stability, also known as D-stability in honor of Germund Dahlquist, refers to the stability of a numerical scheme applied to the simple initial value problem .
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