Vassilios A. Dougalis

On a higher order accurate fully discrete Galerkin approximation to the Navier-Stokes equations

We consider approximating the solution of the initial and boundary value problem for the Navier-Stokes equations in bounded twoand three-dimensional domains using a nonstandard Galerkin (finite element) method for the space discretization and the third order accurate, three-step backward differentiation method (coupled with extrapolation for the nonlinear terms) for the time stepping. The resulting scheme requires the solution of one linear system per time step plus the solution of five linear systems for the computation of the required initial conditions; all these linear systems have the same matrix. The resulting approximations of the velocity are shown to have optimal rate of convergence in L2 under suitable restrictions on the discretization parameters of the problem and the size of the solution in an appropriate function space.

On optimal order error estimates for the nonlinear Schro¨dinger equation

Implicit Runge–Kutta methods in time are used in conjunction with the Galerkin method in space to generate stable and accurate approximations to solutions of the nonlinear (cubic) Schrodinger equation. The temporal component of the discretization error is shown to decrease at the classical rates in some important special cases.

Convergence of Galerkin approximations for the Korteweg-de Vries equation

Standard Galerkin approximations, using smooth splines on a uniform mesh, to 1-periodic solutions of the Korteweg-de Vries equation are analyzed. Optimal rate of convergence estimates are obtained for both semidiscrete and second order in time fully discrete schemes. At each time level, the resulting system of nonlinear equations can be solved by Newtons method. It is shown that if a proper extrapolation is used as a starting value, then only one step of the Newton iteration is required.

Numerical solution of KdV-KdV systems of Boussinesq equations

Considered here is a Boussinesq system of equations from surface water wave theory. The particular system is one of a class of equations derived and analyzed in recent studies. After a brief review of theoretical aspects of this system, attention is turned to numerical methods for the approximation of its solutions with appropriate initial and boundary conditions. Because the system has a spatial structure somewhat like that of the Korteweg-de Vries equation, explicit schemes have unacceptable stability limitations. We instead implement a highly accurate, unconditionally stable scheme that features a Galerkin method with periodic splines to approximate the spatial structure and a two-stage Gauss-Legendre implicit Runge-Kutta method for the temporal discretization. After suitable testing of the numerical scheme, it is used to examine the travelling-wave solutions of the system. These are found to be generalized solitary waves, which are symmetric about their crest and which decay to small amplitude periodic structures as the spatial variable becomes large.

Numerical Approximation of Blow-Up of Radially Symmetric Solutions of the Nonlinear Schrödinger Equation

We consider the initial-value problem for the radially symmetric nonlinear Schrodin\-ger equation with cubic nonlinearity (NLS) in d=2 and 3 space dimensions. To approximate smooth solutions of this problem, we construct and analyze a numerical method based on a standard Galerkin finite element spatial discretization with piecewise linear, continuous functions and on an implicit Crank--Nicolson type time-stepping procedure. We then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to approximate well singular solutions of the NLS equation that blow up at the origin as the temporal variable t tends from below to a finite value

On some high-order accurate fully discrete Galerkin methods for the Korteweg-de Vries equation

t^\star

Numerical solution of Boussinesq systems of KdV-KdV type: II. Evolution of radiating solitary waves

. For the blow-up of the amplitude of the solution we recover numerically the well-known rate

Error estimates for a fully discrete spectral scheme for a class of nonlinear, nonlocal dispersive wave equations

(t^\star - t)^{-1/2}

A Numerical Study of the Stability of Solitary Waves of the Bona-Smith Family of Boussinesq Systems

for d=3. For d=2 our numerical evidence supports the validity of the

Numerical solution of some nonlocal, nonlinear dispersive wave equations

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