High-order unconditionally stable FC-AD solvers for general smooth domains II. Elliptic, parabolic and hyperbolic PDEs; theoretical considerations

Lyon, Mark; Bruno, Oscar P.

doi:10.1016/j.jcp.2010.01.006

Cited by 85 publications

(121 citation statements)

References 27 publications

(50 reference statements)

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“…In the present work, the recently developed Fourier continuation (FC) algorithm and associated efficient parallelization techniques [25][26][27] -which offer essentially spectral ðp ) 1Þ accuracy over the bulk of the domain's interior-are used to produce accurate numerical solutions of the full nonlinear acoustic equations for problems in which such direct simulation was previously impractical or impossible. The method, which can yield accurate solutions using significantly coarser spatial discretization than needed by previous approaches, can lead to improvements in computing times by factors of hundreds or even thousands over those required by other methodologies-thus significantly increasing the feasibility of the direct solution of HIFU problems.…”

Section: Introductionmentioning

confidence: 99%

Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams

Albin

Bruno

Cheung

et al. 2012

The Journal of the Acoustical Society of America

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On the basis of recently developed Fourier continuation (FC) methods and associated efficient parallelization techniques, this text introduces numerical algorithms that, due to very low dispersive errors, can accurately and efficiently solve the types of nonlinear partial differential equation (PDE) models of nonlinear acoustics in hundred-wavelength domains as arise in the simulation of focused medical ultrasound. As demonstrated in the examples presented in this text, the FC approach can be used to produce solutions to nonlinear acoustics PDEs models with significantly reduced discretization requirements over those associated with finite-difference, finite-element and finite-volume methods, especially in cases involving waves that travel distances that are orders of magnitude longer than their respective wavelengths. In these examples, the FC methodology is shown to lead to improvements in computing times by factors of hundreds and even thousands over those required by the standard approaches. A variety of one-and two-dimensional examples presented in this text demonstrate the power and capabilities of the proposed methodology, including an example containing a number of scattering centers and nonlinear multiple-scattering events.

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Section: Introductionmentioning

confidence: 99%

Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams

Albin

Bruno

Cheung

et al. 2012

The Journal of the Acoustical Society of America

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“…Hybridization of that solver with a WENO method, which is left for future work, is expected to give rise to higher efficiencies than the CD6-WENO hybrid. But, in any case, owing to its superior control of pollution error (see Figure 1) and general ability to handle complex, non-periodic domains at high order (as discussed in detail in the introduction sections of references [4,5]), the FC5-WENO9 method is itself expected to be significantly more efficient and flexible than CD6-WENO in applications involving general geometries Table 2 Error in entropy amplification, Mach = 6.0, T = 2.5 N D = 160, N P = 33, ∆t = 2.1 × 10 −4…”

Section: The Euler Systemmentioning

confidence: 99%

“…(Here we restrict our considerations to one dimensional problems. In view of the contributions [1,4,5], which concern FC methods for PDEs in two and three spatial dimensions, however, we expect our FC-WENO approach should be applicable and, indeed, highly competitive for systems of conservation laws in both two-and three-dimensional space.) The FC approximation is based on a high-order periodic continuation of a (possibly nonperiodic) function, yet being a Fourier method, the FC approximation retains the equi-spaced grid points, has no dispersion (pollution) error, and, owing to its reliance on the Fast Fourier Transform, it is highly efficient.…”

Section: Introductionmentioning

confidence: 99%

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Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws

Shahbazi

Albin

Bruno

et al. 2011

Journal of Computational Physics

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We introduce a multi-domain Fourier-Continuation/WENO hybrid method (FC-WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and which enjoys essentially dispersionless, spectral character away from shocks, as well as mild CFL constraints. The hybrid scheme employs the expensive, shock-capturing WENO method in small regions containing discontinuities and the efficient FC method in the rest of the computational domain, yielding a highly effective overall scheme for applications with a mix of discontinuities and complex smooth structures. The accuracy, stability and efficiency of the new FC-based method for conservation laws is investigated for both problems in which solutions are smooth and problems for which solutions contain shock discontinuities; in particular, in the latter case we compare the efficiency of the hybrid FC-WENO method to that of a purely WENO-based approach. In a variety of examples, including an Euler problem that governs the interaction of a strong shock with a very small entropy wave, we show that the hybrid strategy is several times faster than the pure WENO solver for a comparable level of accuracy.

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“…In this work, extending the one-dimensional version of the hybrid approach [39] to two-dimensional problems, we propose an attractive alternative in which we hybridize the WENO method with a recently proposed Fourier continuation (FC) method [1,6,7]. The FC approximation is based on a high-order periodic continuation of a (possibly non-periodic) function, yet being based on a Fourier method, the FC approximation has almost no dispersion (pollution) error and, utilizing an equispaced grid, provides a simple and efficient interface with WENO scheme and the multi-resolution analysis while allowing for an efficient temporal integration as compared to the Chebyshev-WENO scheme.…”

Section: Introductionmentioning

confidence: 99%

Multi-dimensional hybrid Fourier continuation–WENO solvers for conservation laws

Shahbazi

Hesthaven

Zhu

2013

Journal of Computational Physics

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We introduce a multi-dimensional point-wise multi-domain hybrid Fourier-Continuation/WENO technique (FC-WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and essentially dispersionless, spectral, solution away from discontinuities, as well as mild CFL constraints for explicit time stepping schemes. The hybrid scheme conjugates the expensive, shock-capturing WENO method in small regions containing discontinuities with the efficient FC method in the rest of the computational domain, yielding a highly effective overall scheme for applications with a mix of discontinuities and complex smooth structures. The smooth and discontinuous solution regions are distinguished using the multi-resolution procedure of Harten [J. Comput. Phys. 115 (1994) 319-338]. We consider a WENO scheme of formal order nine and a FC method of order five. The accuracy, stability and efficiency of the new hybrid method for conservation laws are investigated for problems with both smooth and non-smooth solutions. The Euler equations for gas dynamics are solved for the Mach 3 and Mach 1.25 shock wave interaction with a small, plain, oblique entropy wave using the hybrid FC-WENO, the pure WENO and the hybrid central difference-WENO (CD-WENO) schemes. We demonstrate considerable computational advantages of the new FC-based method over the two alternatives. Moreover, in solving a challenging two-dimensional Richtmyer-Meshkov instability (RMI), the hybrid solver results in seven-fold speedup over the pure WENO scheme. Thanks to the multi-domain formulation of the solver, the scheme is straightforwardly implemented on parallel processors using message passing interface as well as on Graphics Processing Units (GPUs) using CUDA programming language. The performance of the solver on parallel CPUs yields almost perfect scaling, illustrating the minimal communication requirements of the multi-domain strategy. For the same RMI test, the hybrid computations on a single GPU, in double precision arithmetics, displays five-to six-fold speedup over the hybrid computations on a single CPU. The relative speedup of the hybrid computation over the WENO computations on GPUs is similar to that on CPUs, demonstrating the advantage of hybrid schemes technique on both CPUs and GPUs.

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High-order unconditionally stable FC-AD solvers for general smooth domains II. Elliptic, parabolic and hyperbolic PDEs; theoretical considerations

Cited by 85 publications

References 27 publications

Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams

Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams

Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws

Multi-dimensional hybrid Fourier continuation–WENO solvers for conservation laws

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