A Kernel-Based explicit unconditionally stable scheme for Hamilton-Jacobi equations on nonuniform meshes

Christlieb, Andrew; Sands, William A.; Yang, Hyoseon

doi:10.1016/j.jcp.2020.109543

Cited by 3 publications

(11 citation statements)

References 32 publications

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“…High-order successive convolution algorithms have been developed to solve a range of time-dependent PDEs, including the wave equation [5], heat equation (e.g., Allen-Cahn [6] and Cahn-Hilliard equations [7]), Maxwell's equations [8], Vlasov equation [9], degenerate advection-diffusion (A-D) equation [2], and the Hamilton-Jacobi (H-J) equation [1,3]. In contrast to these papers, this work focuses on the performance of the method in parallel computing environments, which is a largely unexplored area of research.…”

Section: Introductionmentioning

confidence: 99%

“…Recent developments in successive convolution methods have focused on extensions to solve more general nonlinear PDEs, for which an integral solution is generally not applicable. This work considers discretizations developed for degenerate advection-diffusion (A-D) equations [2], as well as the Hamilton-Jacobi (H-J) equations [1,3]. The key idea of these papers exploited the linearity of a given differential operator rather than the underlying equations, allowing derivatives in nonlinear problems to be expressed using the same representations developed for linear problems.…”

Section: Introductionmentioning

confidence: 99%

“…To address shock-capturing and control non-physical oscillations, the latter two papers introduced quadratures that use WENO reconstructions, along with a nonlinear filter to further control oscillations. In [3], the schemes for the H-J equations were extended to enable calculations on mapped grids. This paper also proposed a new WENO quadrature method that uses a basis that consists of exponential polynomials that improves the shock capturing capabilities.…”

Section: Introductionmentioning

confidence: 99%

“…The earlier developments for successive convolution methods, such as [5], are quite similar to Rothe's method in the treatment of the time derivatives. However, successive convolution methods differ from Rothe's method considerably in the treatment of spatial derivatives for nonlinear problems, such as those considered in more recent work on successive convolution (see e.g., [1,2,3]), as Newton iteration can be avoided on nonlinear terms. Additionally, these methods do not require solutions to linear systems.…”

Section: Introductionmentioning

confidence: 99%

“…Despite these advances, multiscale computing problems continue to pose an incredible challenge to modern architectures because they require resolving scales that often vary by orders of magnitude in both space and time. Such complications have led us to consider alternative discretizations for partial differential equations (PDEs) which use expansions involving integral operators to approximate spatial derivatives [1,2,3]. These constructions use explicit information within the integral terms, but treat boundary data implicitly, which contributes to the overall speed of the method.…”

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confidence: 99%

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Parallel Algorithms for Successive Convolution

Christlieb¹,

Guthrey²,

Sands³

et al. 2020

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The development of modern computing architectures with ever-increasing amounts of parallelism has allowed for the solution of previously intractable problems across a variety of scientific disciplines. Despite these advances, multiscale computing problems continue to pose an incredible challenge to modern architectures because they require resolving scales that often vary by orders of magnitude in both space and time. Such complications have led us to consider alternative discretizations for partial differential equations (PDEs) which use expansions involving integral operators to approximate spatial derivatives [1,2,3]. These constructions use explicit information within the integral terms, but treat boundary data implicitly, which contributes to the overall speed of the method. This approach is provably unconditionally stable for linear problems and stability has been demonstrated experimentally for nonlinear problems. Additionally, it is matrix-free in the sense that it is not necessary to invert linear systems and iteration is not required for nonlinear terms. Moreover, the scheme employs a fast summation algorithm that yields a method with a computational complexity of O(N ), where N is the number of mesh points along a coordinate direction. While much work has been done to explore the theory behind these methods, their practicality in large scale computing environments is a largely unexplored topic. In this work, we explore the performance of these methods by developing a domain decomposition algorithm suitable for distributed memory systems along with shared memory algorithms. As a first pass, we derive an artificial Courant-Friedrichs-Lewy (CFL) condition that enforces a nearest-neighbor (N-N) communication pattern and briefly discuss possible generalizations. We also analyze several approaches for implementing the parallel algorithms by optimizing predominant loop structures and maximizing data reuse. Using a hybrid design that employs MPI and Kokkos [4] for the distributed and shared memory components of the algorithms, respectively, we show that our methods are efficient and can sustain an update rate > 1 × 10 8 DOF/node/s. We provide results that demonstrate the scalability and versatility of our algorithms using several different PDE test problems, including a nonlinear example, which employs an adaptive time-stepping rule.

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