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Spectral Methods: Algorithms, Analysis and Applications
Along with finite differences and finite elements, spectral methods are one of the three main methodologies for solving partial differential equations on computers. This book provides a detailed
Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials
  • Jie Shen
  • Mathematics, Computer Science
    SIAM J. Sci. Comput.
  • 1 November 1994
TLDR
This paper presents some efficient algorithms based on the Legendre–Galerkin approximations for the direct solution of the second- and fourth-order elliptic equations using matrix-matrix multiplications for discrete variational formulations.
Efficient Spectral-Galerkin Method II. Direct Solvers of Second- and Fourth-Order Equations Using Chebyshev Polynomials
  • Jie Shen
  • Mathematics, Computer Science
    SIAM J. Sci. Comput.
  • 1995
TLDR
Numerical results indicate that the direct solvers presented in this paper are significantly more accurate and efficient than those based on the Chebyshev-tau method.
A New Class of Efficient and Robust Energy Stable Schemes for Gradient Flows
TLDR
A new numerical technique to deal with nonlinear terms in gradient flows by introducing a scalar auxiliary variable (SAV), which can construct unconditionally second-order energy stable schemes and can easily construct even third or fourth order BDF schemes, although not unconditionally stable, which are very robust in practice.
A Phase-Field Model and Its Numerical Approximation for Two-Phase Incompressible Flows with Different Densities and Viscosities
TLDR
A physically consistent phase-field model that admits an energy law is proposed, and several energy stable, efficient, and accurate time discretization schemes for the coupled nonlinear phase- field model are constructed and analyzed.
On the error estimates for the rotational pressure-correction projection methods
TLDR
The rotational form of the pressure-correction method that was proposed by Timmermans, Minev, and Van De Vosse provides better accuracy in terms of the H1-norm of the velocity and of the L2-normof the pressure than the standard form.
A time-stepping scheme involving constant coefficient matrices for phase-field simulations of two-phase incompressible flows with large density ratios
TLDR
An efficient time-stepping scheme for simulations of the coupled Navier-Stokes Cahn-Hilliard equations for the phase field approach that effectively overcomes the performance bottleneck induced by variable coefficient matrices associated with the variable density and variable viscosity.
On error estimates of projection methods for Navier-Stokes equations: first-order schemes
In this paper projection methods (or fractional step methods) are studied in the semi-discretized form for the Navier–Stokes equations in a two- or three-dimensional bounded domain. Error estimates
An Efficient Spectral-Projection Method for the Navier–Stokes Equations in Cylindrical Geometries: I. Axisymmetric Cases
Abstract An efficient and accurate numerical scheme is presented for the axisymmetric Navier–Stokes equations in primitive variables in a cylinder. The scheme is based on a new spectral-Galerkin
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