A high‐order backward forward sweep interpolating algorithm for semi‐Lagrangian method

Abstract

Conventional semi‐Lagrangian methods often suffer from poor accuracy and imbalance problems of advected properties because of low‐order interpolation schemes used and/or inability to reduce both dissipation and dispersion errors even with high‐order schemes. In the current work, we propose a fourth‐order semi‐Lagrangian method to solve the advection terms at a computing cost of third‐order interpolation scheme by applying backward and forward interpolations in an alternating sweep manner. The method was demonstrated for solving 1‐D and 2‐D advection problems, and 2‐D and 3‐D lid‐driven cavity flows with a multi‐level V‐cycle multigrid solver. It shows that the proposed method can reduce both dissipation and dispersion errors in all regions, especially near sharp gradients, at a same accuracy as but less computing cost than the typical fourth‐order interpolation because of fewer grids used. The proposed method is also shown able to achieve more accurate results on coarser grids than conventional linear and other high‐order interpolation schemes in the literature. Copyright © 2017 John Wiley & Sons, Ltd.

An optimization platform based on coupled indoor environment and HVAC simulation and its application in optimal thermostat placement

“…A Modelica Buildings library (Wetter et al 2014) was developed to facilitate building system modeling. FFD, which simulates airflow and temperature distribution in transient state, is about 50 times faster than its counterpart CFD (Zuo and Chen 2009) and even faster when adopting more advanced semi-Lagrangian algorithm (Mortezazadeh and Wang 2017).…”

SLAC – a semi-Lagrangian artificial compressibility solver for steady-state incompressible flows

“…For this problem, the grid-independent results can be achieved for the grid number beyond 400 Â 400. Meanwhile, the accuracy may be further improved by adopting high-order spatial schemes for the semi-Lagrangian methods as explained in our previous study (Mortezazadeh and Wang, 2017). As the current study focuses on the comparison between SLPB and SLAC methods, we will not dwell on how to improve the simulations, for which readers with interests can refer to our previous study.…”

“…The continuing advancement of computing hardware and the recent booming of data-driven analyses create new opportunities, and thus, challenges of fast and accurate fluid flow solutions for complexed and/or large-scale applications. The semi-Lagrangian method (Courant et al, 1952;Mortezazadeh and Wang, 2017;Wang et al, 2017) of treating advection terms combined with the fractional step method (FSM) (Mortezazadeh and Wang, 2017) for considering the velocity-pressure coupling, offer many advantages including superior stability and natural parallelization when compared to conventional fluid flow solvers. It has, thus, been widely used and will continue to be applied to more complexed and big-scale problems, such as fluid visualization engines for games and films (Stam, 1999;Fedkiw et al, 2001), numerical weather SLAC for steady-state incompressible flows forecasting and ocean flows (Blazek, 2019;Staniforth et al, 1991;Priestley, 1993;Carfora, 2001), multiphase flows (Enright et al, 2002;Wang et al, 2012), building aerodynamics (Zuo et al, 2012;Jin et al, 2013;Jin and Chen, 2015) and electromagnetic applications (Dolean et al, 2010;Grote and Mitkova, 2012).…”

“…For example, Zerroukat et al (2004Zerroukat et al ( , 2005 and Zerroukat (2010) used conservative remapping for the semi-Lagrangian method. Other researchers used high-order, modified and/or hybrid interpolation schemes to reduce dissipation and diffusion interpolating errors, including the monotonic cubic interpolation scheme by Fedkiw et al (2001); a constrained interpolation profile from Song et al (2005); a hybrid combination of low and high-order polynomial interpolation schemes by Zuo et al (2012); a second-order temporal scheme to solve the time derivation term by Xiu and Karniadakis (2001); a combined standard semi-Lagrangian and Crank-Nicolson scheme by Spiegelman and Katz (2006); a second-order scheme in both space and time to reduce numerical viscosity by Kim et al (2007); and a fourth-order backward and forward sweep interpolating algorithm by Mortezazadeh and Wang (2017). On the other hand, to increase the computing speed hindered by poor converging rate due to solving the Poisson equation, various techniques have also been developed.…”

“…The interpolation scheme for the semi-Lagrangian method is the linear interpolation to reduce the computational cost. However, the proposed method is not limited to a linear interpolation scheme but applicable for high-order schemes (Mortezazadeh and Wang, 2017). Additionally, the advection term and also the pressure domain in the proposed method are solved by using a parallel programming code.…”