Abstract: The Wavelet-Galerkin finite element method for
solving the one-dimensional heat equation is presented in this work.
Two types of basis functions which are the Lagrange and multi-level
wavelet bases are employed to derive the full form of matrix system.
We consider both linear and quadratic bases in the Galerkin method.
Time derivative is approximated by polynomial time basis that
provides easily extend the order of approximation in time space. Our
numerical results show that the rate of convergences for the linear
Lagrange and the linear wavelet bases are the same and in order 2
while the rate of convergences for the quadratic Lagrange and the
quadratic wavelet bases are approximately in order 4. It also reveals
that the wavelet basis provides an easy treatment to improve
numerical resolutions that can be done by increasing just its desired
levels in the multilevel construction process.
Abstract: The POD-assisted projective integration method based on the equation-free framework is presented in this paper. The method is essentially based on the slow manifold governing of given system. We have applied two variants which are the “on-line" and “off-line" methods for solving the one-dimensional viscous Bergers- equation. For the on-line method, we have computed the slow manifold by extracting the POD modes and used them on-the-fly along the projective integration process without assuming knowledge of the underlying slow manifold. In contrast, the underlying slow manifold must be computed prior to the projective integration process for the off-line method. The projective step is performed by the forward Euler method. Numerical experiments show that for the case of nonperiodic system, the on-line method is more efficient than the off-line method. Besides, the online approach is more realistic when apply the POD-assisted projective integration method to solve any systems. The critical value of the projective time step which directly limits the efficiency of both methods is also shown.