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Abstract: A systematic approach is presented for constructing higher-order immersed boundary and ghost fluid methods for CFD in general, and fluid–structure interaction problems in particular. Such methods are gaining popularity because they simplify a number of computational issues. These range from gridding the fluid domain, to designing and implementing Eulerian-based algorithms for challenging fluid–structure applications characterized by large structural motions and deformations or topological changes. However, because they typically operate on non body-fitted grids, immersed boundary and ghost fluid methods also complicate other issues such as the treatment of wall boundary conditions in general, and fluid–structure transmission conditions in particular. These methods also tend to be at best first-order space-accurate at the immersed interfaces. In some cases, they are also provably inconsistent at these locations. A methodology is presented in this paper for addressing this issue. It is developed for inviscid flows and prescribed structural motions. For the sake of clarity, but without any loss of generality, this methodology is described in one and two dimensions. However, its extensions to flow-induced structural motions and three dimensions are straightforward. The proposed methodology leads to a departure from the current practice of populating ghost fluid values independently from the chosen spatial discretization scheme. Instead, it accounts for the pattern and properties of a preferred higher-order discretization scheme, and attributes ghost values as to preserve the formal order of spatial accuracy of this scheme. It is illustrated in this paper by its application to various finite difference and finite volume methods. Its impact is also demonstrated by one- and two-dimensional numerical experiments that confirm its theoretically proven ability to preserve higher-order spatial accuracy, including in the vicinity of the immersed interfaces. [Copyright &y& Elsevier]

In this paper, some numerical algorithms (spectral collocation method, block spectral collocation method, boundary value method, block boundary value method, implicit Runge–Kutta method, diagonally implicit Runge–Kutta method and total variation diminishing Runge–Kutta method) are used to solve the highly oscillatory second-order initial value problems. We first derive these methods for the first-order initial value problems, and then extend these methods to the highly oscillatory nonlinear systems by matrix analysis methods. These new methods preserve the accuracy of the original methods and the main advantages of these new methods are low storage requirements and high efficiency. Extensive numerical results are presented to demonstrate the convergence properties of these methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present a new algorithm based on Cartesian meshes for the numerical approximation of kinetic models set in an arbitrary geometry. Due to the high dimensional property of kinetic models, numerical algorithms based on unstructured meshes are not really appropriate since most of numerical methods (semi-Lagrangian, spectral methods) are particularly efficient on structured grids. Here we propose to adapt the inverse Lax–Wendroff procedure, which has been recently introduced for conservation laws [21], for kinetic equations. Numerical simulations in and based on this approach are proposed for Boltzmann type operators (BGK, ES-BGK models). [Copyright &y& Elsevier]

Abstract: In this paper, we derive a series of local structure-preserving algorithms for the “good” Boussinesq equation, including multisymplectic geometric structure-preserving algorithms, local energy-preserving algorithms and local momentum-preserving algorithms. The outstanding advantage of the proposed algorithms is that they conserve these local structures in any time-space region exactly. For example, the proposed local energy-preserving algorithms preserve the local energy conservation law in any local domain. Therefore, the local structure-preserving algorithms overcome the shortage of global structure-preserving algorithms on the boundary conditions. Especially, with suitable boundary conditions such as periodic or homogeneous boundary conditions, the local structure-preserving algorithms will be global structure-preserving algorithms. Numerical results verify the theoretical analysis. [Copyright &y& Elsevier]

Abstract: This paper is concerned with spectral Galerkin algorithms for solving high even-order two point boundary value problems in one dimension subject to homogeneous and nonhomogeneous boundary conditions. The proposed algorithms are extended to solve two-dimensional high even-order differential equations. The key to the efficiency of these algorithms is to construct compact combinations of Chebyshev polynomials of the third and fourth kinds as basis functions. The algorithms lead to linear systems with specially structured matrices that can be efficiently inverted. Numerical examples are included to demonstrate the validity and applicability of the proposed algorithms, and some comparisons with some other methods are made. [Copyright &y& Elsevier]

Abstract: This paper deals with a new solid–fluid coupling algorithm between a rigid body and an unsteady compressible fluid flow, using an Embedded Boundary method. The coupling with a rigid body is a first step towards the coupling with a Discrete Element method. The flow is computed using a finite volume approach on a Cartesian grid. The expression of numerical fluxes does not affect the general coupling algorithm and we use a one-step high-order scheme proposed by Daru and Tenaud [V. Daru, C. Tenaud, J. Comput. Phys. (2004)]. The Embedded Boundary method is used to integrate the presence of a solid boundary in the fluid. The coupling algorithm is totally explicit and ensures exact mass conservation and a balance of momentum and energy between the fluid and the solid. It is shown that the scheme preserves uniform movement of both fluid and solid and introduces no numerical boundary roughness. The efficiency of the method is demonstrated on challenging one- and two-dimensional benchmarks. [Copyright &y& Elsevier]

Abstract: Common efficient schemes for the incompressible Navier–Stokes equations, such as projection or fractional step methods, have limited temporal accuracy as a result of matrix splitting errors, or introduce errors near the domain boundaries (which destroy uniform convergence to the solution). In this paper we recast the incompressible (constant density) Navier–Stokes equations (with the velocity prescribed at the boundary) as an equivalent system, for the primary variables velocity and pressure. equation for the pressure. The key difference from the usual approaches occurs at the boundaries, where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time. This avoids the common difficulty of an, apparently, over-determined Poisson problem. Since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity, the recast equations are ideal for numerical marching methods. The new system can be discretized using a variety of methods, including semi-implicit treatments of viscosity, and in principle to any desired order of accuracy. In this work we illustrate the approach with a 2-D second order finite difference scheme on a Cartesian grid, and devise an algorithm to solve the equations on domains with curved (non-conforming) boundaries, including a case with a non-trivial topology (a circular obstruction inside the domain). This algorithm achieves second order accuracy in the L ∞ norm, for both the velocity and the pressure. The scheme has a natural extension to 3-D. [Copyright &y& Elsevier]

Abstract: Typical VOF algorithms rely on an implicit slip that scales with mesh refinement, to allow contact lines to move along no-slip boundaries. As a result, solutions of contact line phenomena vary continuously with mesh spacing; this paper presents examples of that variation. A mesh-dependent dynamic contact angle model is then presented, that is based on fundamental hydrodynamics and serves as a more appropriate boundary condition at a moving contact line. This new boundary condition eliminates the stress singularity at the contact line; the resulting problem is thus well-posed and yields solutions that converge with mesh refinement. Numerical results are presented of a solid plate withdrawing from a fluid pool, and of spontaneous droplet spread at small capillary and Reynolds numbers. [Copyright &y& Elsevier]

Abstract: In this paper we propose a numerical method for computing minimal surfaces with fixed boundaries. The level set method is used to evolve a codimension-1 surface with fixed codimension-2 boundary in under mean curvature flow. For n =3 the problem has been approached in D.L. Chopp, 1993 and L.-T. Cheng [D.L. Chopp, Computing minimal surfaces via level set curvature flow, J. Comput. Phys. 106(1) (1993) 77–91 and L.-T. Cheng, The level set method applied to geometrically based motion, materials science, and image processing, UCLA CAM Report, 00-20] using the level set method, but with a more complicated boundary conditions. The method we present can be generalized straightforward to arbitrary dimension, and the framework in which it is presented is dimension independent. Examples are shown for n =2, 3, 4. [Copyright &y& Elsevier]