
Parallel implicitexplicit general linear methods
Highorder discretizations of partial differential equations (PDEs) nece...
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High order semiimplicit multistep methods for time dependent partial differential equations
We consider the construction of semiimplicit linear multistep methods w...
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Fast parallel solution of fully implicit RungeKutta and discontinuous Galerkin in time for numerical PDEs, Part I: the linear setting
Fully implicit RungeKutta (IRK) methods have many desirable properties ...
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Linearly Stabilized Schemes for the Time Integration of Stiff Nonlinear PDEs
In many applications, the governing PDE to be solved numerically contain...
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Fast parallel solution of fully implicit RungeKutta and discontinuous Galerkin in time for numerical PDEs, Part II: nonlinearities and DAEs
Fully implicit RungeKutta (IRK) methods have many desirable accuracy an...
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Multirate LinearlyImplicit GARK Schemes
Many complex applications require the solution of initialvalue problems...
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Domain Decomposition for the Closest Point Method
The discretization of elliptic PDEs leads to large coupled systems of eq...
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High order linearly implicit methods for evolution equations: How to solve an ODE by inverting only linear systems
This paper introduces a new class of numerical methods for the time integration of evolution equations set as Cauchy problems of ODEs or PDEs. The systematic design of these methods mixes the RungeKutta collocation formalism with collocation techniques, in such a way that the methods are linearly implicit and have high order. The fact that these methods are implicit allows to avoid CFL conditions when the large systems to integrate come from the space discretization of evolution PDEs. Moreover, these methods are expected to be efficient since they only require to solve one linear system of equations at each time step, and efficient techniques from the literature can be used to do so. After the introduction of the methods, we set suitable definitions of consistency and stability for these methods. This allows for a proof that arbitrarily high order linearly implicit methods exist and converge when applied to ODEs. Eventually, we perform numerical experiments on ODEs and PDEs that illustrate our theoretical results for ODEs, and compare our methods with standard methods for several evolution PDEs.
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