Related papers: High-order asynchrony-tolerant finite difference s…
Synchronization overheads pose a major challenge as applications advance towards extreme scales. In current large-scale algorithms, synchronization as well as data communication delay the parallel computations at each time step in a…
We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for…
Next-generation exascale machines with extreme levels of parallelism will provide massive computing resources for large scale numerical simulations of complex physical systems at unprecedented parameter ranges. However, novel numerical…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
Direct numerical simulations (DNS) are an indispensable tool for understanding the fundamental physics of turbulent flows. Because of their steep increase in computational cost with Reynolds number ($R_{\lambda}$), well-resolved DNS are…
We introduce a novel class of finite difference approximations, termed zigzag schemes, that employ a hybrid stencil that is neither symmetrical, nor fully one-sided. These zigzag schemes often enjoy more permissive stability constraints and…
Stencil composition uses the idea of function composition, wherein two stencils with arbitrary orders of derivative are composed to obtain a stencil with a derivative order equal to sum of the orders of the composing stencils. In this…
Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in…
To overcome the communication bottlenecks observed in state-of-the-art parallel time-dependent flow solvers at extreme scales, an asynchronous computing approach that relaxes communication and synchronization at a mathematical level was…
Symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented. They have the same three or four-dimensional symmetry groups as the original differential equations. The new…
In this work we explore the fidelity of numerical approximations to the analytic spectra of hyperbolic partial differential equation systems with variable coefficients. We are particularly interested in the ability of discrete methods to…
We present a new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time. Our scheme combines asymptotic techniques (which are inexpensive but can have insufficient accuracy) with parallel-in-time methods (which,…
We propose a new method to design adaptation algorithms that guarantee a certain prescribed level of performance and are applicable to systems with nonconvex parameterization. The main idea behind the method is, given the desired…
Within recent years, considerable progress has been made regarding high-performance solvers for Partial Differential Equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely…
This paper introduces an adaptive time splitting technique for the solution of stiff evolutionary PDEs that guarantees an effective error control of the simulation, independent of the fastest physical time scale for highly unsteady…
For the parallel computation of partial differential equations, one key is the grid partitioning. It requires that each process owns the same amount of computations, and also, the partitioning quality should be proper to reduce the…
In this paper we discuss three symbolic approaches for the generation of a finite difference scheme of a partial differential equation (PDE). We prove, that for a linear PDE with constant coefficients these three approaches are equivalent…
The need to smoothly cover a computational domain of interest generically requires the adoption of several grids. To solve the problem of interest under this grid-structure one must ensure the suitable transfer of information among the…
Propagation characteristics of a wave are defined by the dispersion relationship, from which the governing partial differential equation (PDE) can be recovered. PDEs are commonly solved numerically using the finite-difference (FD) method,…
When solving partial differential equations using classical schemes such as finite difference or finite volume methods, sufficiently fine meshes and carefully designed schemes are required to achieve high-order accuracy of numerical…