SIAM Journal on Numerical Analysis
Inertial manifolds and multigrid methods
SIAM Journal on Mathematical Analysis
Nonlinear Galerkin methods: the finite elements case
Numerische Mathematik
Incremental unknowns for solving partial differential equations
Numerische Mathematik
Construction of approximate inertial manifolds using wavelets
SIAM Journal on Mathematical Analysis
Solution of generalized Stokes problems using hierarchical methods and incremental unknowns
Applied Numerical Mathematics
A nonlinear Galerkin method: the two-level Fourier-collocation case
Journal of Scientific Computing
Applied Numerical Mathematics
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Time Marching Multilevel Techniques for Evolutionary Dissipative Problems
SIAM Journal on Scientific Computing
Accurate Computations on Inertial Manifolds
SIAM Journal on Scientific Computing
Multilevel schemes for the shallow water equations
Journal of Computational Physics
Incremental unknowns method based on the θ-scheme for time-dependent convection-diffusion equations
Mathematics and Computers in Simulation
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In this article, we conjugate time marching schemes with Finite Differences splittings into low and high modes in order to build fully explicit methods with enhanced temporal stability for the numerical solutions of PDEs. The main idea is to apply explicit schemes with less restrictive stability conditions to the linear term of the high modes equation, in order that the allowed time step for the temporal integration is only determined by the low modes. These conjugated schemes were developed in [10] for the spectral case and here we adapt them to the Finite Differences splittings provided by Incremental Unknowns, which steems from the Inertial Manifolds theory. We illustrate their improved capabilities with numerical solutions of Burgers equations, with uniform and nonuniform meshes, in dimensions one and two, when using modified Forward–Euler and Adams–Bashforth schemes. The resulting schemes use time steps of the same order of those used by semi-implicit schemes with comparable accuracy and reduced computational costs.