Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
On Krylov Subspace Approximations to the Matrix Exponential Operator
SIAM Journal on Numerical Analysis
An exponential method of numerical integration of ordinary differential equations
Communications of the ACM
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
High-Order Stiff ODE Solvers via Automatic Differentiation and Rational Prediction
WNAA '96 Proceedings of the First International Workshop on Numerical Analysis and Its Applications
The Scaling and Squaring Method for the Matrix Exponential Revisited
SIAM Journal on Matrix Analysis and Applications
Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation
Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation
Exponential Rosenbrock-Type Methods
SIAM Journal on Numerical Analysis
VSVO formulation of the taylor method for the numerical solution of ODEs
Computers & Mathematics with Applications
Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators
SIAM Journal on Scientific Computing
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For the time integration of semilinear systems of differential equations, a class of multiderivative exponential integrators is considered. The methods are based on a Taylor series expansion of the semilinearity about the numerical solution, the required derivatives are computed by automatic differentiation. Inserting these derivatives into the variation-of-constants formula results in an exponential integrator which requires the action of the exponential of an augmented Jacobian only. The convergence properties of such exponential integrators are analyzed, and potential sources of numerical instabilities are identified. In particular, it is shown that local linearization gives rise to better stability for stiff problems. A number of numerical experiments illustrate the theoretical results.