57 research outputs found

    SpECTRE: A Task-based Discontinuous Galerkin Code for Relativistic Astrophysics

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    We introduce a new relativistic astrophysics code, SpECTRE, that combines a discontinuous Galerkin method with a task-based parallelism model. SpECTRE's goal is to achieve more accurate solutions for challenging relativistic astrophysics problems such as core-collapse supernovae and binary neutron star mergers. The robustness of the discontinuous Galerkin method allows for the use of high-resolution shock capturing methods in regions where (relativistic) shocks are found, while exploiting high-order accuracy in smooth regions. A task-based parallelism model allows efficient use of the largest supercomputers for problems with a heterogeneous workload over disparate spatial and temporal scales. We argue that the locality and algorithmic structure of discontinuous Galerkin methods will exhibit good scalability within a task-based parallelism framework. We demonstrate the code on a wide variety of challenging benchmark problems in (non)-relativistic (magneto)-hydrodynamics. We demonstrate the code's scalability including its strong scaling on the NCSA Blue Waters supercomputer up to the machine's full capacity of 22,380 nodes using 671,400 threads.Comment: 41 pages, 13 figures, and 7 tables. Ancillary data contains simulation input file

    Galerkin approximations for the optimal control of nonlinear delay differential equations

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    Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042

    Numerical Methods for Hyperbolic Partial Differential Equations

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    Department of Mathematical SciencesIn this dissertation, new numerical methods are proposed for different types of hyperbolic partial differential equations (PDEs). The objectives of these developments aim for the improvements in accuracy, robustness, efficiency, and reduction of the computational cost. The dissertation consists of two parts. The first half discusses shock-capturing methods for nonlinear hyperbolic conservation laws, and proposes a new adaptive weighted essentially non-oscillatory WENO-?? scheme in the context of finite difference. Depending on the smoothness of the large stencils used in the reconstruction of the numerical flux, a parameter ?? is set adaptively to switch the scheme between a 5th-order upwind or 6th-order central discretization. A new indicator depending on parameter ?? measures the smoothness of the large stencils in order to choose a smoother one for the reconstruction procedure. ?? is devised based on the possible highest-order variations of the reconstructing polynomials in an L2 sense. In addition, a new set of smoothness indicators ??_k???s of the sub-stencils is introduced. These are constructed in a central sense with respect to the Taylor expansions around point x_j . Numerical results show that the new scheme combines good properties of both 5th-order upwind and 6th-order central schemes. In particular, the new scheme captures discontinuities and resolves small-scaled structures much better than other 5th-order schemes; overcomes the loss of resolution near some critical regions; and is able to maintain symmetry which are drawbacks detected in other 6th-order central WENO schemes. The second part extends the scope to hyperbolic PDEs with uncertainty, and semi-analytical methods using singular perturbation analysis for dispersive PDEs. For the former, a hybrid operator splitting method is developed for computation of the two-dimensional transverse magnetic Maxwell equations in media with multiple random interfaces. By projecting the solutions into random space using the Polynomial Chaos (PC) expansions, the deterministic and random parts of the solution are solved separately. The deterministic parts are then numerically approximated by the FDTD method with domain decomposition implemented on a staggered grid. Statistic quantities are obtained by the Monte Carlo sampling in the post-processing stage. Parallel computing is proposed for which the computational cost grows linearly with the number of random interfaces. The last section deals with spectral methods for dispersive PDEs. The Kortewegde Vries (KdV) equation is chosen as a prototype. By Fourier series, the PDE is transformed into a system of ODEs which is stiff, that is, there are rapid oscillatory modes for large wavenumbers. A new semi-analytical method is proposed to tackle the difficulty. The new method is based on the classical integrating factor (IF) and exponential time differencing (ETD) schemes. The idea is to approximate analytically the stiff parts by the so-called correctors and numerically the non-stiff parts by the IF and ETD methods. It turns out that rapid oscillations are well absorbed by our corrector method, yielding better accuracy in the numerical results. Due to the nonlinearity, all Fourier modes interact with each other, causing the computation of the correctors to be very costly. In order to overcome this, the correctors are recursively constructed to accurately capture the stiffness of the mode interactions.ope

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

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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 discontinuities, as well as mild CFL constraints (comparable to those of finite difference methods). 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 smooth and discontinuous solution regions are distinguished using the multi-resolution procedure of Harten [J. Comput. Phys. 115 (1994) 319-338]. We consider WENO schemes of formal orders five and nine and a FC method of order five. The accuracy, stability and efficiency of the new hybrid method for conservation laws is investigated for problems with both smooth and non-smooth solutions. In the latter case, we solve the Euler equations for gas dynamics for the standard test case of a Mach three shock wave interacting with an entropy wave, as well as a shock wave (with Mach 1.25, three or six) interacting with a very small entropy wave and evaluate the efficiency of the hybrid FC-WENO method as compared to a purely WENO-based approach as well as alternative hybrid based techniques. We demonstrate considerable computational advantages of the new FC-based method, suggesting a potential of an order of magnitude acceleration over alternatives when extended to fully three-dimensional problems. (C) 2011 Elsevier Inc. All rights reserved

    Higher-Order Finite Difference Scheme With TVD Filter Algorithm. Part 1: Simulations of Scalar Advective Dominant Problems.

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    Abstract Engquist et al. (1989) proposed non-linear TVD filters. When combined with a traditional higher-order finite difference scheme, these filters can simulate shock or high concentration gradient problems with no spurious oscillations. Among the filter proposed is the filter algorithm 2.2. However, this algorithm flattens extrema that are not the results of overshooting and consequency the scheme reduces to a low order of accuracy locally around smooth extrema. Modification of the TVD filter algorithm 2.2 has been proposed in this paper to overcome this problem. Several conservative finite difference schemes are considered for testing the TVD filter.  Non-conservative schemes consisting of 4th and 6th -order Runge-Kutta method are also evaluated. The modified filter has been tested to simulate seven test cases, including a pure advection of scalar profiles, a pure advection with variable velocity, two inviscid burger equations, an advection-diffusion equation with variable velocity and dispersion, advection of three solid bodies in rotating fluid around a square of a side length of 2, and a two-dimensional advection-diffusion equation. The numerical experiments showed that applying the modified TVD filter, combined with the higher-order non-TVD finite difference schemes for solving the advection equation, can produce accurate solutions with no oscillations and no clipping effect extrema. Keywords : Non-Oscillatory schemes, TVD filter, advection, higher-order accurate Abstrak Engquist dkk. (1989) mengusulkan filter TVD non-linear. Ketika dikombinasikan dengan skema beda hingga orde tinggi tradisional, filter ini dapat mensimulasikan kejutan atau masalah gradien konsentrasi tinggi tanpa osilasi palsu. Di antara filter yang diusulkan adalah algoritma filter 2.2. Namun, algoritma ini meratakan ekstrem yang bukan merupakan hasil dari “overshooting” dan konsekuensinya skema tersebut direduksi ke tingkat akurasi yang rendah secara lokal di sekitar ekstrem halus. Modifikasi algoritma filter TVD 2.2 telah diusulkan dalam makalah ini untuk mengatasi masalah ini. Beberapa skema konservatif dipertimbangkan untuk menguji filter TVD. Skema non-konservatif metode Runge-Kutta orde ke-4 dan ke-6 juga dievaluasi. Filter yang dimodifikasi diuji untuk mensimulasikan tujuh kasus, termasuk adveksi murni profil skalar, adveksi murni dengan kecepatan bervariasi, dua persamaan burger inviscid, persamaan adveksi-difusi dengan kecepatan bervariasi dan dispersi, adveksi tiga benda padat berputar oleh aliran di bidang persegi dengan panjang sisi 2, dan persamaan adveksi-difusi dua dimensi. Eksperimen numerik menunjukkan bahwa penerapan filter TVD yang dimodifikasi, dikombinasikan dengan skema beda hingga non-TVD orde tinggi untuk menyelesaikan persamaan adveksi, dapat menghasilkan solusi yang akurat tanpa osilasi dan tanpa efek kliping Kata Kunci : skema tanpa osilasi, Filter TVD, advection, orde-akurasi tingg
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