A Parameterized multi-step Newton method for solving systems of nonlinear equations

We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter. to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of theta, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spatial and temporal independent variables such as, for instance, the Chebyshev pseudo-spectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radius of convergence than the multi-step Newton method.Peer ReviewedPostprint (author's final draft

A multidomain spectral method for solving elliptic equations

We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three distinct features. First, the combined problem of solving the PDE, satisfying the boundary conditions, and matching between different subdomains is cast into one set of equations readily accessible to standard linear and nonlinear solvers. Second, touching as well as overlapping subdomains are supported; both rectangular blocks with Chebyshev basis functions as well as spherical shells with an expansion in spherical harmonics are implemented. Third, the code is very flexible: The domain decomposition as well as the distribution of collocation points in each domain can be chosen at run time, and the solver is easily adaptable to new PDEs. The code has been used to solve the equations of the initial value problem of general relativity and should be useful in many other problems. We compare the new method to finite difference codes and find it superior in both runtime and accuracy, at least for the smooth problems considered here.Comment: 31 pages, 8 figure

The automatic solution of partial differential equations using a global spectral method

A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential operators and the one-dimensional ultraspherical spectral method. If a partial differential operator is of splitting rank $2$, such as the operator associated with Poisson or Helmholtz, the corresponding PDE is solved via a generalized Sylvester matrix equation, and a bivariate polynomial approximation of the solution of degree $(n_x,n_y)$ is computed in $\mathcal{O}((n_x n_y)^{3/2})$ operations. Partial differential operators of splitting rank $\geq 3$ are solved via a linear system involving a block-banded matrix in $\mathcal{O}(\min(n_x^{3} n_y,n_x n_y^{3}))$ operations. Numerical examples demonstrate the applicability of our 2D spectral method to a broad class of PDEs, which includes elliptic and dispersive time-evolution equations. The resulting PDE solver is written in MATLAB and is publicly available as part of CHEBFUN. It can resolve solutions requiring over a million degrees of freedom in under $60$ seconds. An experimental implementation in the Julia language can currently perform the same solve in $10$ seconds.Comment: 22 page

Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains

Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate spectral methods using rational basis (or modified mapped Gegenbauer functions) for such models in unbounded domains. The main building block of the spectral algorithms is the explicit representations for the Fourier transform and fractional Laplacian of the rational basis, derived from some useful integral identites related to modified Bessel functions. With these at our disposal, we can construct rational spectral-Galerkin and direct collocation schemes by pre-computing the associated fractional differentiation matrices. We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces, and analyze the optimal convergence of the proposed Galerkin scheme. We also provide ample numerical results to show that the rational method outperforms the Hermite function approach

Spectral methods in general relativistic astrophysics

We present spectral methods developed in our group to solve three-dimensional partial differential equations. The emphasis is put on equations arising from astrophysical problems in the framework of general relativity.Comment: 51 pages, elsart (Elsevier Preprint), 19 PostScript figures, submitted to Journal of Computational & Applied Mathematic

A Hybrid Radial Basis Function - Pseudospectral Method for Thermal Convection in a 3-D Spherical Shell

A novel hybrid spectral method that combines radial basis function (RBF) and Chebyshev pseudospectral (PS) methods in a “2+1” approach is presented for numerically simulating thermal convection in a 3-D spherical shell. This is the first study to apply RBFs to a full 3D physical model in spherical geometry. In addition to being spectrally accurate, RBFs are not defined in terms of any surface based coordinate system such as spherical coordinates. As a result, when used in the lateral directions, as in this study, they completely circumvent the pole issue with the further advantage that nodes can be “scattered” over the surface of a sphere. In the radial direction, Chebyshev polynomials are used, which are also spectrally accurate and provide the necessary clustering near the boundaries to resolve boundary layers. Applications of this new hybrid methodology are given to the problem of convection in the Earth’s mantle,which is modeled by a Boussinesq fluid at infinite Prandtl number. To see whether this numerical technique warrants further investigation, the study limits itself to an isoviscous mantle.Benchmark comparisons are presented with other currently used mantle convection codes for Rayleigh number 7 · 103 and 105. The algorithmic simplicity of the code (mostly due to RBFs)allows it to be written in less than 400 lines of Matlab and run on a single workstation. We find that our method is very competitive with those currently used in the literature