On a new explicit time-integration method for advection equation and its application in hydrodynamics

Yanlin Shao*

*Corresponding author for this work

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Abstract

The advection equation is a differential equation that governs the motion of a conserved scalar field as it is advected by a known velocity vector field. It has high relevance in solution of problems related to water waves and their interaction with fixed and oating structures. When the seakeeping problem with forward speed effects is considered in a reference frame moving with the ship speed, advective terms appear in both kinematic and dynamic free surface conditions [1] [2]. The advection terms are also an important part of the Navier-Stokes equations, representing great numerical challenges in terms of both accuracy and stability for ows dominated by advection. There exist many numerical methods that can be used to solve the advection equation in the time domain. Examples are the one-stage explict and implict Euler methods, as well as multi-stage methods which can be built from the one-stage methods. Explicit methods are normally easy to implement and cheaper to solve within one time step for a given spatial discretization. However, they normally require small time steps to achieve stable solution if possible. The stability of the explicit methods is strongly affected by the numerical scheme for the spatial discretization. It can be shown by a von Neumann stability analysis that, if a scheme based on forward in time and central difference in spatial derivatives is used in solving a periodic problem, the solution is unconditionally unstable. Therefore, upwind finite difference schemes are often applied to stabilize the solution, and the Courant{Friedrichs{Lewy (CFL) number must be smaller enough. We present a new class of explicit scheme which is derived from an approximation of the implicit Euler scheme. This scheme does not involve solving matrix equations that are required by a standard implicit Euler scheme. Unlike the standard explicit Euler scheme, which is unconditionally unstable for central difference schemes in spatial descretization, the proposed scheme is conditionally stable for any types of differential operators in the advection terms. The linear stability analysis results and two examples of the application of the new explicit schemes will also be shown.
Original languageEnglish
Publication date2020
Number of pages4
Publication statusPublished - 2020
Event35th International Workshop on Water Waves and Floating Bodies (IWWWFB 2020) - Seoul National University, Soul, Korea, Republic of
Duration: 26 Apr 202029 Apr 2020

Conference

Conference35th International Workshop on Water Waves and Floating Bodies (IWWWFB 2020)
LocationSeoul National University
CountryKorea, Republic of
CitySoul
Period26/04/202029/04/2020

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