Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries

Abstract

The diffuse-domain, or smoothed boundary, method is an attractive approach for solving partial differential equations in complex geometries because of its simplicity and flexibility. In this method the complex geometry is embedded into a larger, regular domain. The original PDE is reformulated using a smoothed characteristic function of the complex domain and source terms are introduced to approximate the boundary conditions. The reformulated equation, which is independent of the dimension and domain geometry, can be solved by standard numerical methods and the same solver can be used for any domain geometry. A challenge is making the method higher-order accurate. For Dirichlet boundary conditions, which we focus on here, current implementations demonstrate a wide range in their accuracy but generally the methods yield at best first order accuracy in , the parameter that characterizes the width of the region over which the characteristic function is smoothed. Typically, ∝ h, the grid size. Here, we analyze the diffuse-domain PDEs using matched asymptotic expansions and explain the observed behaviors. Our analysis also identifies simple modifications to the diffuse-domain PDEs that yield higher-order accuracy in , e.g., O( 2 ) in the L 2 norm and O( p ) with 1.5 ≤ p ≤ 2 in the L ∞ norm. Our analytic results are confirmed numerically in stationary and moving domains where the level set method is used to capture the dynamics of the domain boundary and to construct the smoothed characteristic function.

A diffuse domain method for two-phase flows with large density ratio in complex geometries

“…The numerical set-up is identical to that considered above. We find that the errors are of a similar order of magnitude but are larger than those using BC u = − −2 (1 − φ)(u − u g ) and are sub-first-order accurate, which is consistent with the results of Yu et al (2020). In the following, we use BC u = − −2 (1 − φ)(u − u g ) exclusively.…”

“…This is likely due to the vector form of the problem, and the quasi-incompressibility condition, which couples the derivatives of the velocity. In future work, we will extend the asymptotic analysis in § 4 and use the approach developed in Yu et al (2020) to guide the development of a higher-order-accurate version of the q-NSCH-DD system. Moreover, we will also consider dynamic contact angle formulations developed in Liu & Wu (2019).…”

“…Remark. Another choice of BC u = − −3 (1 − φ)(u − u g ) was presented by Li et al (2009) and later the scheme was shown to be O( ln( )) accurate by Yu et al (2020). We here provide the numerical results using BC u = − −3 (1 − φ)(u − u g ) for the cavity flow example in tables 3 and 4.…”

“…The original partial differential equation is then reformulated on the larger domain with additional source terms that approximate the boundary conditions. It has been shown that the reformulated partial differential equation asymptotically converges to the original system as the width of the diffuse interface layer tends to zero (Li et al 2009;Yu, Chen & Thornton 2012;Lervag & Lowengrub 2015;Chadwick et al 2018;Poulsen & Voorhees 2018;Yu, Guo & Lowengrub 2020). The advantages of such a method are that no modifications of standard software packages are required and the stationary/moving complex boundary can be imposed easily and smoothly.…”

“…2009; Yu, Chen & Thornton 2012; Lervag & Lowengrub 2015; Chadwick et al. 2018; Poulsen & Voorhees 2018; Yu, Guo & Lowengrub 2020). The advantages of such a method are that no modifications of standard software packages are required and the stationary/moving complex boundary can be imposed easily and smoothly.…”