Implicit Multiderivative Collocation Solvers for Linear Partial Differential Equations with Discontinuous Galerkin Spatial Discretizations

In this work, we construct novel discretizations for the unsteady convection–diffusion equation. Our discretization relies on multiderivative time integrators together with a novel discretization that reduces the total number of unkowns for the solver. These type of temporal discretizations come fro...

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Bibliographic Details
Published in:Journal of scientific computing 2017-12, Vol.73 (2-3), p.1145-1163
Main Authors: Schütz, Jochen, Seal, David C., Jaust, Alexander
Format: Article
Language:English
Subjects:
Accuracy
Algorithms
Collocation methods
Computational Mathematics and Numerical Analysis
Convection-diffusion equation
Discretization
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Methods
Ordinary differential equations
Partial differential equations
Runge-Kutta method
Solvers
Theoretical
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Summary:In this work, we construct novel discretizations for the unsteady convection–diffusion equation. Our discretization relies on multiderivative time integrators together with a novel discretization that reduces the total number of unkowns for the solver. These type of temporal discretizations come from an umbrella class of methods that include Lax–Wendroff (Taylor) as well as Runge–Kutta methods as special cases. We include two-point collocation methods with multiple time derivatives as well as a sixth-order fully implicit collocation method that only requires a total of three stages. Numerical results for a number of sample linear problems indicate the expected order of accuracy and indicate we can take arbitrarily large time steps.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-017-0485-9