Topic
Finite difference
About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.
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TL;DR: In this paper, the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems is discussed, and three possibilities are investigated, their O(h2)-convergence established and illustrated by numerical examples.
Abstract: We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (x?y?)?=f(x,y), y(0)=A, y(1)=B, 0
93 citations
TL;DR: Numerical comparisons with pure finite-difference methods demonstrate the effectiveness of techniques that combine grid and particle solvers for the solution of the incompressible Navier--Stokes equations for various flow geometries, bounded or unbounded.
Abstract: We describe and illustrate numerical procedures that combine grid and particle solvers for the solution of the incompressible Navier--Stokes equations. These procedures include vortex in cell (VIC) and domain decomposition schemes. Numerical comparisons with pure finite-difference methods demonstrate the effectiveness of these techniques for various flow geometries, bounded or unbounded.
93 citations
TL;DR: Sharp lower bounds are obtained for multiplications and storage in the sparse system arising from the application of finite difference or finite element techniques to linear boundary value problems on plane regions yielding regular $n \times n$ grids.
Abstract: Sharp lower bounds are obtained for multiplications and storage in the sparse system arising from the application of finite difference or finite element techniques to linear boundary value problems on plane regions yielding regular $n \times n$ grids. Graph-theoretic techniques are used to take advantage of the simplicity of the underlying combinatorial structure of the problem.
93 citations
TL;DR: In this paper, a novel discrete boundary condition for wide angle parabolic equations (WAPEs) is derived from the fully discretized whole-space problem that is reflection-free and yields an unconditionally stable scheme.
93 citations
TL;DR: A hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions is described, and how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space is shown.
Abstract: We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. We simulate the algorithm to solve an instance of Navier-Stokes equations and compute density, temperature, and velocity profiles for the fluid flow in a convergent-divergent nozzle.
93 citations