71,343 research outputs found
An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations
In this paper, we consider the initial boundary value problem of the two
dimensional multi-term time fractional mixed diffusion and diffusion-wave
equations. An alternating direction implicit (ADI) spectral method is developed
based on Legendre spectral approximation in space and finite difference
discretization in time. Numerical stability and convergence of the schemes are
proved, the optimal error is , where are the
polynomial degree, time step size and the regularity of the exact solution,
respectively. We also consider the non-smooth solution case by adding some
correction terms. Numerical experiments are presented to confirm our
theoretical analysis. These techniques can be used to model diffusion and
transport of viscoelastic non-Newtonian fluids
Nodally exact Ritz discretizations of 1D diffusion–absorption and Helmholtz equations by variational FIC and modified equation methods
The final publication is available at Springer via http://dx.doi.org/10.1007/s00466-005-0011-zThis article presents the first application of the Finite Calculus (FIC) in a Ritz-FEM variational framework. FIC provides a steplength parametrization of mesh dimensions, which is used to modify the shape functions. This approach is applied to the FEM discretization of the steady-state, one-dimensional, diffusion–absorption and Helmholtz equations. Parametrized linear shape functions are directly inserted into a FIC functional. The resulting Ritz-FIC equations are symmetric and carry a element-level free parameter coming from the function modification process. Both constant- and variable-coefficient cases are studied. It is shown that the parameter can be used to produce nodally exact solutions for the constant coefficient case. The optimal value is found by matching the finite-order modified differential equation (FOMoDE) of the Ritz-FIC equations with the original field equation. The inclusion of the Ritz-FIC models in the context of templates is examined. This inclusion shows that there is an infinite number of nodally exact models for the constant coefficient case. The ingredients of these methods (FIC, Ritz, MoDE and templates) can be extended to multiple dimensions.Peer ReviewedPostprint (author's final draft
Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution
We present a new solver for nonlinear parabolic problems that is L-stable and
achieves high order accuracy in space and time. The solver is built by first
constructing a single-dimensional heat equation solver that uses fast O(N)
convolution. This fundamental solver has arbitrary order of accuracy in space,
and is based on the use of the Green's function to invert a modified Helmholtz
equation. Higher orders of accuracy in time are then constructed through a
novel technique known as successive convolution (or resolvent expansions).
These resolvent expansions facilitate our proofs of stability and convergence,
and permit us to construct schemes that have provable stiff decay. The
multi-dimensional solver is built by repeated application of dimensionally
split independent fundamental solvers. Finally, we solve nonlinear parabolic
problems by using the integrating factor method, where we apply the basic
scheme to invert linear terms (that look like a heat equation), and make use of
Hermite-Birkhoff interpolants to integrate the remaining nonlinear terms. Our
solver is applied to several linear and nonlinear equations including heat,
Allen-Cahn, and the Fitzhugh-Nagumo system of equations in one and two
dimensions
Matrix-equation-based strategies for convection-diffusion equations
We are interested in the numerical solution of nonsymmetric linear systems
arising from the discretization of convection-diffusion partial differential
equations with separable coefficients and dominant convection. Preconditioners
based on the matrix equation formulation of the problem are proposed, which
naturally approximate the original discretized problem. For certain types of
convection coefficients, we show that the explicit solution of the matrix
equation can effectively replace the linear system solution. Numerical
experiments with data stemming from two and three dimensional problems are
reported, illustrating the potential of the proposed methodology
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