A Boundary Element Method for Multiple Moving Boundary Problems

Zerroukat, M.; Wrobel, L.C.

doi:10.1006/jcph.1997.5829

Cited by 10 publications

(1 citation statement)

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“…The work of [2,20] is one of the first attempts to design fast methods for the heat equation. A direct formulation was used in [10,28] for the 1D Stefan problem. In the case of prescribed Dirichlet data, a direct integral equation formulation leads to a Volterra system of equations that is ill-conditioned.…”

Section: The Main Contributions Of This Work Are the High-order Time-mentioning

confidence: 99%

A High-Order Solver for the Heat Equation in 1D domains with Moving Boundaries

Veerapaneni

Biros

2007

SIAM J. Sci. Comput.

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We describe a fast high-order accurate method for the solution of the heat equation in domains with moving Dirichlet or Neumann boundaries and distributed forces. We assume that the motion of the boundary is prescribed. Our method extends the work of Greengard and Strain [Comm. Pure Appl. Math., XLIII (1990), pp. 949-963]. Our scheme is based on a time-space Chebyshev pseudo-spectral collocation discretization, which is combined with a recursive product quadrature rule to accurately and efficiently approximate convolutions with Green's function for the heat equation. We present numerical results that exhibit up to eighth-order convergence rates. Assuming N time steps and M spatial discretization points, the evaluation of the solution of the heat equation at the same number of points in space-time requires O(N M log M) work. Thus, our scheme can be characterized as "fast"; that is, it is work-optimal up to a logarithmic factor. Abstract. We describe a fast high-order accurate method for the solution of the heat equation in domains with moving Dirichlet or Neumann boundaries and distributed forces. We assume that the motion of the boundary is prescribed. Our method extends the work of Greengard and Strain [Comm. Pure Appl. Math., XLIII (1990), pp. 949-963]. Our scheme is based on a time-space Chebyshev pseudo-spectral collocation discretization, which is combined with a recursive product quadrature rule to accurately and efficiently approximate convolutions with Green's function for the heat equation. We present numerical results that exhibit up to eighth-order convergence rates. Assuming N time steps and M spatial discretization points, the evaluation of the solution of the heat equation at the same number of points in space-time requires O(NM log M ) work. Thus, our scheme can be characterized as "fast"; that is, it is work-optimal up to a logarithmic factor.

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Section: The Main Contributions Of This Work Are the High-order Time-mentioning

confidence: 99%

A High-Order Solver for the Heat Equation in 1D domains with Moving Boundaries

Veerapaneni

Biros

2007

SIAM J. Sci. Comput.

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Evaluation of 2‐D Green's boundary formula and its normal derivative using Legendre polynomials, with an application to acoustic scattering problems

Yang

2001

Numerical Meth Engineering

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SUMMARYThis paper presents the non-singular forms, in a global sense, of two-dimensional Green's boundary formula and its normal derivative. The main advantage of the modiÿed formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain; that is, the proposed element-free method requires only nodal data. The approach includes expressing the unknown function as a truncated Fourier-Legendre series, together with transforming the integration interval [a; b] to [−1; 1]; the series coe cients are thus to be determined. The hypersingular integral, interpreted in the Hadamard ÿnite-part sense, and some weakly singular integrals can be evaluated analytically; the remaining integrals are regular with the limiting values of the integrands deÿned explicitly when a source point coincides with a ÿeld point. The e ectiveness of the modiÿed formulations is examined by an elliptic cylinder subject to prescribed boundary conditions.The regularization is further applied to acoustic scattering problems. The well-known Burton-Miller method, using a linear combination of the surface Helmholtz integral equation and its normal derivative, is adopted to overcome the non-uniqueness problem. A general non-singular form of the composite equation is derived. Comparisons with analytical solutions for acoustically soft and hard circular cylinders are made.

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