Multi‐stage high order semi‐Lagrangian schemes for incompressible flows in Cartesian geometries

Efficient transport algorithms are essential to the numerical resolution of incompressible fluid‐flow problems. Semi‐Lagrangian methods are widely used in grid based methods to achieve this aim. The accuracy of the interpolation strategy then determines the properties of the scheme. We introduce a simple multi‐stage procedure, which can easily be used to increase the order of accuracy of a code based on multilinear interpolations. This approach is an extension of a corrective algorithm introduced by Dupont & Liu (2003, 2007). This multi‐stage procedure can be easily implemented in existing parallel codes using a domain decomposition strategy, as the communication pattern is identical to that of the multilinear scheme. We show how a combination of a forward and backward error correction can provide a third‐order accurate scheme, thus significantly reducing diffusive effects while retaining a non‐dispersive leading error term. Copyright © 2016 John Wiley & Sons, Ltd.     

A semi‐Lagrangian level set method for incompressible Navier–Stokes equations with free surface

In this paper, we formulate a level set method in the framework of finite elements‐semi‐Lagrangian methods to compute the solution of the incompressible Navier–Stokes equations with free surface. In our formulation, we use a quasi‐monotone semi‐Lagrangian scheme, which is both unconditionally stable and essentially non oscillatory, to compute the advective terms in the Navier–Stokes equations, the transport equation and the equation of the reinitialization stage for the level set function. The method we propose is quite robust and flexible with regard to the mesh and the geometry of the domain, as well as the magnitude of the Reynolds number. We illustrate the performance of the method in several examples, which range from a benchmark problem to test the volume conservation property of the method to the flow past a NACA0012 foil at high Reynolds number. Copyright © 2005 John Wiley & Sons, Ltd.     

On the efficiency of semi‐implicit and semi‐Lagrangian spectral methods for the calculation of incompressible flows

Classical semi‐implicit backward Euler/Adams–Bashforth time discretizations of the Navier–Stokes equations induce, for high‐Reynolds number flows, severe restrictions on the time step. Such restrictions can be relaxed by using semi‐Lagrangian schemes essentially based on splitting the full problem into an explicit transport step and an implicit diffusion step. In comparison with the standard characteristics method, the semi‐Lagrangian method has the advantage of being much less CPU time consuming where spectral methods are concerned. This paper is devoted to the comparison of the ‘semi‐implicit’ and ‘semi‐Lagrangian’ approaches, in terms of stability, accuracy and computational efficiency. Numerical results on the advection equation, Burger's equation and finally two‐ and three‐dimensional Navier–Stokes equations, using spectral elements or a collocation method, are provided. Copyright © 2001 John Wiley & Sons, Ltd.     

A semi‐Lagrangian multi‐moment finite volume method with fourth‐order WENO projection

Numerical oscillation has been an open problem for high‐order numerical methods with increased local degrees of freedom (DOFs). Current strategies mainly follow the limiting projections derived originally for conventional finite volume methods and thus are not able to make full use of the sub‐cell information available in the local high‐order reconstructions. This paper presents a novel algorithm that introduces a nodal value‐based weighted essentially non‐oscillatory limiter for constrained interpolation profile/multi‐moment finite volume method (CIP/MM FVM) (Ii and Xiao, J. Comput. Phys., 222 (2007), 849–871) as an effort to pursue a better suited formulation to implement the limiting projection in schemes with local DOFs. The new scheme, CIP‐CSL‐WENO4 scheme, extends the CIP/MM FVM method by limiting the slope constraint in the interpolation function using the weighted essentially non‐oscillatory (WENO) reconstruction that makes use of the sub‐cell information available from the local DOFs and is built from the point values at the solution points within three neighboring cells, thus resulting a more compact WENO stencil. The proposed WENO limiter matches well the original CIP/MM FVM, which leads to a new scheme of high accuracy, algorithmic simplicity, and computational efficiency. We present the numerical results of benchmark tests for both scalar and Euler conservation laws to manifest the fourth‐order accuracy and oscillation‐suppressing property of the proposed scheme. Copyright © 2016 John Wiley & Sons, Ltd.     

Semi‐Lagrangian semi‐implicit time‐splitting scheme for the shallow water equations

Time‐splitting technique applied in the context of the semi‐Lagrangian semi‐implicit method allows the use of extended time steps mainly based on physical considerations and reduces the number of numerical operations at each time step such that it is approximately proportional to the number of the points of spatial grid. To control time growth of the additional truncation errors, the standard stabilizing correction method is modified with no penalty for accuracy and efficiency of the algorithm. A linear analysis shows that constructed scheme is stable for time steps up to 2h. Numerical integrations with actual atmospheric fields of pressure and wind confirm computational efficiency, extended stability and accuracy of the proposed scheme. Copyright © 2006 John Wiley & Sons, Ltd.     

An unconditionally stable semi‐Lagrangian method for the spherical atmospherical shallow water equations

A semi‐implicit, semi‐Lagrangian, mixed finite difference–finite volume model for the shallow water equations on a rotating sphere is introduced and discussed. Its main features are the vectorial treatment of the momentum equation and the finite volume approach for the continuity equation. Pressure and Coriolis terms in the momentum equation and velocity in the continuity equation are treated semi‐implicitly. Moreover, a splitting technique is introduced to preserve symmetry of the numerical scheme. An alternative asymmetric scheme (without splitting) is also introduced and the efficiency of both is discussed. The model is shown to be conservative in geopotential height and unconditionally stable for 0.5≤θ≤1. Numerical experiments on two standard test problems confirm the performance of the model. Copyright © 2000 John Wiley & Sons, Ltd.     

High‐order 𝒞1 finite‐element interpolating schemes—Part II: Nonlinear semi‐Lagrangian shallow‐water models

The finite‐element, semi‐implicit, and semi‐Lagrangian methods are used on unstructured meshes to solve the nonlinear shallow‐water system. Several ??¹ approximation schemes are developed for an accurate treatment of the advection terms. The employed finite‐element discretization schemes are the P–P₁ and P₂–P₁ pairs. Triangular finite elements are attractive because of their flexibility for representing irregular boundaries and for local mesh refinement. By tracking the characteristics backward from both the interpolation and quadrature nodes and using ??¹ interpolating schemes, an accurate treatment of the nonlinear terms and, hence, of Rossby waves is obtained. Results of test problems to simulate slowly propagating Rossby modes illustrate the promise of the proposed approach in ocean modelling. Copyright © 2007 John Wiley & Sons, Ltd.     

Parallel implementation of a non‐hydrostatic model for free surface flows with semi‐Lagrangian advection treatment

The parallel implementation of an unstructured‐grid, three‐dimensional, semi‐implicit finite difference and finite volume model for the free surface Navier–Stokes equations (UnTRIM ) is presented and discussed. The new developments are aimed to make the code available for high‐performance computing in order to address larger, complex problems in environmental free surface flows. The parallelization is based on the mesh partitioning method and message passing and has been achieved without negatively affecting any of the advantageous properties of the serial code, such as its robustness, accuracy and efficiency. The key issue is a new, autonomous parallel streamline backtracking algorithm, which allows using semi‐Lagrangian methods in decomposed meshes without compromising the scalability of the code. The implementation has been carefully verified not only with simple, abstract test cases illustrating the application domain of the code but also with advanced, high‐resolution models presently applied for research and engineering projects. The scheme performance and accuracy aspects are researched and discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

High accuracy time and space transform method for advection–diffusion equation in an unbounded domain

A new numerical method called high accuracy time and space transform method (TSTM) is introduced to solve the advection–diffusion equation in an unbounded domain. By a spatial transform, the advection–diffusion equation in the unbounded domain Rⁿ is converted to one on the bounded domain [?1, 1]ⁿ, and the Laplace transform is applied to eliminate time dependency. The consequent boundary value problem is solved by collocation on Chebyshev points. To face the well‐known computational challenge represented by the numerical inversion of the Laplace transform, Talbot's method is applied, consisting of numerically integrating the Bromwich integral on a special contour by means of trapezoidal or midpoint rules. Numerical experiments illustrate that TSTM has exponential rate in time and space. Copyright © 2008 John Wiley & Sons, Ltd.     

An efficient method for capturing free boundaries in multi‐fluid simulations

An easy‐to‐use front capturing method is devised by directly solving the transport equation for a volume of fluid (VOF) function. The key to this method is a semi‐Lagrangian conservative scheme, namely CIP_CSL3, recently proposed by the author. In the CIP_CSL3 scheme, the first‐order derivative of the interpolation polynomial at each cell centre is used to control the shape of the reconstructed profile. We show in the present paper that the first‐order derivative, which plays a crucial role in reconstructing the interpolation profile, can also be used to eliminate numerical diffusion. The resulting algorithm can be directly used to compute the VOF‐like function and retain the compact thickness of the moving interface in multi‐fluid simulations. No surface reconstruction based on the value of VOF function is required in the method, which makes it quite economical and easy to use. The presented method has been tested with various interfacial flows including pure rotation, vortex shearing, multi‐vortex deformation and the moving boundaries in real fluid as well. The method gives promising results to all computed problems. Copyright © 2003 John Wiley & Sons, Ltd.     

Modified Lagrangian vortex method with improved boundary conditions for water waves past a thin bottom‐standing barrier

Herein, the modified Lagrangian vortex method (LVM), a hybrid analytical‐numerical algorithm per se, is devised to simulate the process of vortex formation and shedding from the sharp edge of a zero‐thickness vertical plate under linear water‐wave attack. Application of the Helmholtz decomposition facilitates a convenient switch between the inviscid‐ and viscous‐flow models, thereby enabling easy incorporation of vorticity effects into the potential‐flow calculations for the viscous‐dominated region. In evaluating the potential‐flow component, making good use of the quickly convergent technique with singular basis functions, correctly capturing the singular behavior in velocity fields near the tip of the plate, leads to a considerable reduction of computational burdens and to 12‐decimal‐place accuracy. The viscous correction is carried out via the meshless LVM with improved boundary conditions. Comparisons with previously published results show good agreement. Simulations of vortex generation and evolution illuminate the ability of the present method, and provide a supplement to pertinent experimental works. The hybrid scheme proposed herein allows flexibility for the former LVM and convenience in the code development. Such a compromise fits particularly well for the high‐resolution modeling of sharp‐edged vortex shedding without heavy numerical developments. Copyright © 2014 John Wiley & Sons, Ltd.     

Insights on a sign‐preserving numerical method for the advection–diffusion equation

In this paper we explore theoretically and numerically the application of the advection transport algorithm introduced by Smolarkiewicz to the one‐dimensional unsteady advection–diffusion equation. The scheme consists of a sequence of upwind iterations, where the initial iteration is the first‐order accurate upwind scheme, while the subsequent iterations are designed to compensate for the truncation error of the preceding step. Two versions of the method are discussed. One, the classical version of the method, regards the second‐order terms of the truncation error and the other considers additionally the third‐order terms. Stability and convergence are discussed and the theoretical considerations are illustrated through numerical tests. The numerical tests will also indicate in which situations are advantageous to use the numerical methods presented. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical modelling of advection–dispersion equation in a stretched curvilinear grid using the QUICKEST scheme

A stretched version of the QUICKEST scheme for solutions of the advection–dispersion equation is presented. The scheme is accurate for large degrees of stretching, so that it can be used where large gradients are present, e.g. for the calculation of sediment in suspension close to the bed. The scheme is tested for various cases of sediment advection and dispersion in one and two dimensions. © 1998 John Wiley & Sons, Ltd.     

An efficient semi‐implicit subgrid method for free‐surface flows on hierarchical grids

We present a new modelling strategy for improving the efficiency of computationally intensive flow problems in environmental free‐surface flows. The approach combines a recently developed semi‐implicit subgrid method with a hierarchical grid solution strategy. The method allows the incorporation of high‐resolution data on subgrid scale to obtain a more accurate and efficient hydrodynamic model. The subgrid method improves the efficiency of the hierarchical grid method by providing better solutions on coarse grids. The method is applicable to both steady and unsteady flows, but we particularly focus on river flows with steady boundary conditions. There, the combined hierarchical grid–subgrid method reduces the computational effort to obtain a steady state with factors up to 43. For unsteady models, the method can be used for efficiently generating accurate initial conditions on high‐resolution grids. Additionally, the method provides automatic insight in grid convergence. We demonstrate the efficiency and applicability of the method using a schematic test for the vortex shedding around a circular cylinder and a real‐world river case study. Copyright © 2015 John Wiley & Sons, Ltd.     

A novel interface‐tracking method based on Lagrangian particles for deformation analysis of a red blood cell in a capillary

A novel efficient interface‐tracking method is developed to gain an insight into the interface in a multiphase or multifluid system, called the modified particle binary level set (MPBLS) method, in which the binary level set function is defined to distinguish the different phases or fluids and further modified by Lagrangian particles scattered along the interface for achieving higher accuracy. The validation of the MPBLS method is carried out first by simulating the free motion of a red blood cell (RBC) in the rotating, shear and Poiseuille flows, respectively. Subsequently, further validations are performed by comparing with the experimental and numerical results published previously. As one of important applications, the MPBLS method is employed to investigate the deformation behaviors of RBCs with different shapes in a capillary. The simulations show that the healthy RBC gradually changes the geometric shape from a biconcave to a steady parachute shape. It is thus guaranteed that the RBC successfully traverses through the smaller capillaries compared with undeformed RBC. However, the unhealthy RBC with the circular or elliptical shape has different deformation behaviors, in which the steady parachute shape is much less concave at the rear and more convex in the front. Copyright © 2011 John Wiley & Sons, Ltd.     

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection‐dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two‐dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.     

A conservative semi‐implicit method for coupled surface–subsurface flows in regional scale

A semi‐implicit method for coupled surface–subsurface flows in regional scale is proposed and analyzed. The flow domain is assumed to have a small vertical scale as compared with the horizontal extents. Thus, after hydrostatic approximation, the simplified governing equations are derived from the Reynolds averaged Navier–Stokes equations for the surface flow and from the Darcy's law for the subsurface flow. A conservative free‐surface equation is derived from a vertical integral of the incompressibility condition and extends to the whole water column including both, the surface and the subsurface, wet domains. Numerically, the horizontal domain is covered by an unstructured orthogonal grid that may include subgrid specifications. Along the vertical direction a simple z‐layer discretization is adopted. Semi‐implicit finite difference equations for velocities and a finite volume approximation for the free‐surface equation are derived in such a fashion that, after simple manipulation, the resulting discrete free‐surface equation yields a single, well‐posed, mildly nonlinear system. This system is efficiently solved by a nested Newton‐type iterative method that yields simultaneously the pressure and a non‐negative fluid volume throughout the computational grid. The time‐step size is not restricted by stability conditions dictated by friction or surface wave speed. The resulting algorithm is simple, extremely efficient, and very accurate. Exact mass conservation is assured also in presence of wetting and drying dynamics, in pressurized flow conditions, and during free‐surface transition through the interface. A few examples illustrate the model applicability and demonstrate the effectiveness of the proposed algorithm. Copyright © 2015 John Wiley & Sons, Ltd.