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Register a new userImmersed Boundary Smooth Extension (IBSE): A high-order method for solving incompressible flows in arbitrary smooth domains
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The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet for fluid problems it only achieves first-order spatial accuracy near embedded boundaries for the velocity field and fails to converge pointwise for elements of the stress tensor. In a previous work we introduced the Immersed Boundary Smooth Extension (IBSE) method, a variation of the IB method that achieves high-order accuracy for elliptic PDE by smoothly extending the unknown solution of the PDE from a given smooth domain to a larger computational domain, enabling the use of simple Cartesian-grid discretizations. In this work, we extend the IBSE method to allow for the imposition of a divergence constraint, and demonstrate high-order convergence for the Stokes and incompressible Navier-Stokes equations: up to third-order pointwise convergence for the velocity field, and second-order pointwise convergence for all elements of the stress tensor. The method is flexible to the underlying discretization: we demonstrate solutions produced using both a Fourier spectral discretization and a standard second-order finite-difference discretization.
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The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet it only achieves first-order spatial accuracy near embedded boundaries. In this paper, we introduce a new high-order numerical method which we call the Immersed Boundary Smooth Extension (IBSE) method. The IBSE method achieves high-order accuracy by smoothly extending the unknown solution of the PDE from a given smooth domain to a larger computational domain, enabling the use of simple Cartesian-grid discretizations (e.g. Fourier spectral methods). The method preserves much of the flexibility and robustness of the original IB method. In particular, it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta-functions to communicate information between the computational grid and the boundary. We present a fast algorithm for solving elliptic equations, which forms the basis for simple, high-order implicit-time methods for parabolic PDE and implicit-explicit methods for related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat, Burgers, and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise convergence for Dirichlet problems and third-order pointwise convergence for Neumann problems.
The dissolution of solids has created spectacular geomorphologies ranging from centimeter-scale cave scallops to the kilometer-scale stone forests of China and Madagascar. Mathematically, dissolution processes are modeled by a Stefan problem, which describes how the motion of a phase-separating interface depends on local concentration gradients, coupled to a fluid flow. Simulating these problems is challenging, requiring the evolution of a free interface whose motion depends on the normal derivatives of an external field in an ever-changing domain. Moreover, density differences created in the fluid domain induce self-generated convecting flows that further complicate the numerical study of dissolution processes. In this contribution, we present a numerical method for the simulation of the Stefan problem coupled to a fluid flow. The scheme uses the Immersed Boundary Smooth Extension method to solve the bulk advection-diffusion and fluid equations in the complex, evolving geometry, coupled to a {theta}-L scheme that provides stable evolution of the boundary. We demonstrate third-order temporal and pointwise spatial convergence of the scheme for the classical Stefan problem, and second-order temporal and pointwise spatial convergence when coupled to flow. Examples of dissolution of solids that result in high-Rayleigh number convection are numerically studied, and qualitatively reproduce the complex morphologies observed in recent experiments.
In this paper a method is presented for evaluating the convolution of the Greens function for the Laplace operator with a specified function $rho(vec x)$ at all grid points in a rectangular domain $Omega subset {mathrm R}^{d}$ ($d = 1,2,3$), i.e. a solution of Poissons equation in an infinite domain. 4th and 6th ord
The G-equation is a well-known model for studying front propagation in turbulent combustion. In this paper, we develop an efficient model reduction method for computing textcolor{black}{regular solutions} of viscous G-equations in incompressible steady and time-periodic cellular flows. Our method is based on the Galerkin proper orthogonal decomposition (POD) method. To facilitate the algorithm design and convergence analysis, we decompose the solution of the viscous G-equation into a mean-free part and a mean part, where their evolution equations can be derived accordingly. We construct the POD basis from the solution snapshots of the mean-free part. With the POD basis, we can efficiently solve the evolution equation for the mean-free part of the solution to the viscous G-equation. After we get the mean-free part of the solution, the mean of the solution can be recovered. We also provide rigorous convergence analysis for our method. Numerical results for textcolor{black}{viscous G-equations and curvature G-equations} are presented to demonstrate the accuracy and efficiency of the proposed method. In addition, we study the turbulent flame speeds of the viscous G-equations in incompressible cellular flows.
A novel sharp interface ghost-cell based immersed boundary method has been proposed and its parameters have been optimized against an analytical model in diffusion applications. The proposed embedded constrained moving least-squares (ECMLS) algorithm minimizes the error of the interpolated concentration at the image point of the ghost point by applying a moving least-squares method on all internal nodes, near the ghost image point, and the associated mirrored image points of these internal nodes through the corresponding boundary conditions. Using an analytical model as a reference, the ECMLS algorithm is compared to the constrained moving least-squares (CMLS) algorithm and the staircase model using various grid sizes, interpolation basis functions, weight functions, and the penalty parameter of the constraint. It is found that using ECMLS algorithm in the investigated diffusion application, the incomplete quartic basis function yields the best performance while the quadratic, cubic, and bicubic basis functions also produce results better than the staircase model. It is also found that the linear and bilinear basis functions cannot produce results better than the staircase model in diffusion applications. It is shown that the optimal radius of the region of internal nodes used for interpolation scales with the logarithm of the boundary radius of curvature. It is shown that for the diffusion application, the proposed ECMLS algorithm produces lower errors at the boundary with better numerical stability over a wider range of basis functions, weight functions, boundary radius of curvature, and the penalty parameter than the CMLS algorithm.
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