7 research outputs found
Orthogonal polynomials on a class of planar algebraic curves
We construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form ym = Ï(x) in R2 where m = 1, 2 and Ï is a polynomial of arbitrary degree d, in terms of univariate semiclassical OPs. We compute connection coefficients that relate the bivariate OPs to a polynomial basis that is itself orthogonal and whose span contains the OPs as a subspace. The connection matrix is shown to be banded and the connection coefficients and Jacobi matrices for OPs of degree 0, . . . , N are computed via the Lanczos algorithm in O(Nd4) operations
Blow up in a periodic semilinear heat equation
Blow up in a one-dimensional semilinear heat equation is studied using a combination of numerical and analytical tools. The focus is on problems periodic in the space variable and starting out from a nearly flat, positive initial condition. Novel results include asymptotic approximations of the solution on different timescales that are, in combination, valid over the entire space and time interval right up to and including the blow-up time. Both the asymptotic analysis and the numerical methods benefit from a well-known reciprocal substitution that transforms the problem into one that does not blow up but remains bounded. This allows for highly accurate computations of blow-up times and the solution profile at the critical time, which are then used to confirm the asymptotics. The approach also makes it possible to continue a solution numerically beyond the singularity. The specific post-blow-up dynamics are believed to be presented here for the first time
Multidomain spectral method for the Gauss hypergeometric function
We present a multidomain spectral approach for Fuchsian ordinary differential equations in the particular case of the hypergeometric equation. Our hybrid approach uses Frobeniusâ method and Moebius transformations in the vicinity of each of the singular points of the hypergeometric equation, which leads to a natural decomposition of the real axis into domains. In each domain, solutions to the hypergeometric equation are constructed via the well-conditioned ultraspherical spectral method. The solutions are matched at the domain boundaries to lead to a solution which is analytic on the whole compactified real line RâȘâ , except for the singular points and cuts of the Riemann surface on which the solution is defined. The solution is further extended to the whole Riemann sphere by using the same approach for ellipses enclosing the singularities. The hypergeometric equation is solved on the ellipses with the boundary data from the real axis. This solution is continued as a harmonic function to the interior of the disk by solving the Laplace equation in polar coordinates with an optimal complexity Fourierâultraspherical spectral method. In cases where logarithms appear in the solution, a hybrid approach involving an analytical treatment of the logarithmic terms is applied. We show for several examples that machine precision can be reached for a wide class of parameters, but also discuss almost degenerate cases where this is not possible
Spurious Resonance in SemiDiscrete Methods for the Korteweg--de Vries Equation
A multiple scales analysis of semidiscrete methods for the Korteweg--de Vries equation is conducted. Methods that approximate the spatial derivatives by finite differences with arbitrary order accuracy and the limiting method, the Fourier pseudospectral method, are considered. The analysis reveals that a resonance effect can occur in the semidiscrete solution but not in the solution of the continuous equation. It is shown for the Fourier pseudospectral discretization that resonance can only be caused by aliased modes. The spurious semidiscrete solutions are investigated in numerical experiments and we suggest methods for avoiding spurious resonance
Orthogonal polynomials on planar cubic curves
Orthogonal polynomials in two variables on cubic curves are considered. For an integral with respect to an appropriate weight function defined on a cubic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. We show that these orthogonal polynomials can be used to approximate functions with cubic and square root singularities, and demonstrate their usage for solving differential equations with singular solutions
Blow up in a periodic semilinear heat equation
Blow up in a one-dimensional semilinear heat equation is studied using a combination of numerical and analytical tools. The focus is on problems periodic in the space variable and starting out from a nearly flat, positive initial condition. Novel results include asymptotic approximations of the solution on different timescales that are, in combination, valid over the entire space and time interval right up to and including the blow-up time. Both the asymptotic analysis and the numerical methods benefit from a well-known reciprocal substitution that transforms the problem into one that does not blow up but remains bounded. This allows for highly accurate computations of blow-up times and the solution profile at the critical time, which are then used to confirm the asymptotics. The approach also makes it possible to continue a solution numerically beyond the singularity. The specific post-blow-up dynamics are believed to be presented here for the first time
Quadratic Pade approximation : numerical aspects and applications
CITATION: Fasondini, M. et al. 2019. Quadratic Pade approximation : numerical aspects and applications. Computer Research and Modeling, 11(6):1017â1031, doi:10.20537/2076-7633-2019-11-6-1017-1031.The original publication is available at http://www.mathnet.ruPadÂŽe approximation is a useful tool for extracting singularity information from a power series. A linear
PadÂŽe approximant is a rational function and can provide estimates of pole and zero locations in the complex
plane. A quadratic PadÂŽe approximant has square root singularities and can, therefore, provide additional
information such as estimates of branch point locations. In this paper, we discuss numerical aspects of computing
quadratic PadÂŽe approximants as well as some applications. Two algorithms for computing the coefficients in
the approximant are discussed: a direct method involving the solution of a linear system (well-known in the
mathematics community) and a recursive method (well-known in the physics community). We compare the
accuracy of these two methods when implemented in floating-point arithmetic and discuss their pros and cons.
In addition, we extend Lukeâs perturbation analysis of linear PadÂŽe approximation to the quadratic case and
identify the problem of spurious branch points in the quadratic approximant, which can cause a significant loss
of accuracy. A possible remedy for this problem is suggested by noting that these troublesome points can be
identified by the recursive method mentioned above. Another complication with the quadratic approximant arises
in choosing the appropriate branch. One possibility, which is to base this choice on the linear approximant, is
discussed in connection with an example due to Stahl. It is also known that the quadratic method is capable of
providing reasonable approximations on secondary sheets of the Riemann surface, a fact we illustrate here by
means of an example. Two concluding applications show the superiority of the quadratic approximant over its
linear counterpart: one involving a special function (the Lambert W-function) and the other a nonlinear PDE
(the continuation of a solution of the inviscid Burgers equation into the complex plane).Publisher's versio