A non‐iterative implicit algorithm for the solution of advection–diffusion equation on a sphere

A numerical algorithm for the solution of advection–diffusion equation on the surface of a sphere is suggested. The velocity field on a sphere is assumed to be known and non‐divergent. The discretization of advection–diffusion equation in space is carried out with the help of the finite volume method, and the Gauss theorem is applied to each grid cell. For the discretization in time, the symmetrized double‐cycle componentwise splitting method and the Crank–Nicolson scheme are used. The numerical scheme is of second order approximation in space and time, correctly describes the balance of mass of substance in the forced and dissipative discrete system and is unconditionally stable. In the absence of external forcing and dissipation, the total mass and L₂‐norm of solution of discrete system is conserved in time. The one‐dimensional periodic problems arising at splitting in the longitudinal direction are solved with Sherman–Morrison's formula and Thomas's algorithm. The one‐dimensional problems arising at splitting in the latitudinal direction are solved by the bordering method that requires a prior determination of the solution at the poles. The resulting linear systems have tridiagonal matrices and are solved by Thomas's algorithm. The suggested method is direct (without iterations) and rapid in realization. It can also be applied to linear and nonlinear diffusion problems, some elliptic problems and adjoint advection–diffusion problems on a sphere. Copyright © 2015 John Wiley & Sons, Ltd.     

Efficient preconditioning techniques for finite‐element quadratic discretization arising from linearized incompressible Navier–Stokes equations

We develop an efficient preconditioning techniques for the solution of large linearized stationary and non‐stationary incompressible Navier–Stokes equations. These equations are linearized by the Picard and Newton methods, and linear extrapolation schemes in the non‐stationary case. The time discretization procedure uses the Gear scheme and the second‐order Taylor–Hood element P₂?P₁ is used for the approximation of the velocity and the pressure. Our purpose is to develop an efficient preconditioner for saddle point systems. Our tools are the addition of stabilization (penalization) term r?(div(·)), and the use of triangular block matrix as global preconditioner. This preconditioner involves the solution of two subsystems associated, respectively, with the velocity and the pressure and have to be solved efficiently. Furthermore, we use the P₁?P₂ hierarchical preconditioner recently proposed by the authors, for the block matrix associated with the velocity and an additive approach for the Schur complement approximation. Finally, several numerical examples illustrating the good performance of the preconditioning techniques are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

An efficient and accurate fully discrete finite element method for unsteady incompressible Oldroyd fluids with large time step

This paper proposes a second‐order accuracy in time fully discrete finite element method for the Oldroyd fluids of order one. This new approach is based on a finite element approximation for the space discretization, the Crank–Nicolson/Adams–Bashforth scheme for the time discretization and the trapezoid rule for the integral term discretization. It reduces the nonlinear equations to almost unconditionally stable and convergent systems of linear equations that can be solved efficiently and accurately. Here, the numerical simulations for L², H¹ error estimates of the velocity and L² error estimates of the pressure at different values of viscoelastic viscosities α, different values of relaxation time λ₁, different values of null viscosity coefficient μ₀ are shown. In addition, two benchmark problems of Oldroyd fluids with different solvent viscosity μ and different relaxation time λ₁ are simulated. All numerical results perfectly match with the theoretical analysis and show that the developed approach gives a high accuracy to simulate the Oldroyd fluids under a large time step. Furthermore, the difference and the connection between the Newton fluids and the viscoelastic Oldroyd fluids are displayed. Copyright © 2015 John Wiley & Sons, Ltd.     

A design of residual error estimates for a high order BDF‐DGFE method applied to compressible flows

We deal with the numerical solution of the non‐stationary compressible Navier–Stokes equations with the aid of the backward difference formula – discontinuous Galerkin finite element method. This scheme is sufficiently stable, efficient and accurate with respect to the space as well as time coordinates. The nonlinear algebraic systems arising from the backward difference formula – discontinuous Galerkin finite element discretization are solved by an iterative Newton‐like method. The main benefit of this paper are residual error estimates that are able to identify the computational errors following from the space and time discretizations and from the inexact solution of the nonlinear algebraic systems. Thus, we propose an efficient algorithm where the algebraic, spatial and temporal errors are balanced. The computational performance of the proposed method is demonstrated by a list of numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

On the critical problem of F. D. pressure treatment for laminar flows confined by permeable walls

In the present work the viscous (low Reynolds) flow in plane ducts confined by permeable walls has been studied. A simple model of the filtrating walls has been used, with the normal velocity component proportional to the pressure jump across the wall, resulting in a non-standard boundary value Navier-Stokes problem. A critical analysis of the appropriate boundary condition and pressure problem has led to the conclusions of employing a simple explicit finite volume approach, and of avoiding the use of higher order finite difference schemes. In this paper a special emphasis on the structure of the involved computational matrices has been given to illustrate the chosen algorithm. The latter yields a steady state solution that is second order accurate in space, and it has an accuracy in time of order ≤ Δt (the time step), due to the explicit treatment of the velocity boundary conditions along the membrane. The model has been tested to study the effects of the inlet/outlet conditions, Reynolds number and filtrating wall constant.     

Verification of deforming polarized structure computation by using a closed-form solution

Various engineering systems exploit the conversion between electromagnetic and mechanical work. It is important to compute this coupling accurately, and we present a method for solving the governing equations simultaneously (at once) without a staggering scheme. We briefly present the theory for coupling the elecgoverning equations as well as the variational formulation that leads to the weak form. This weak form is nonlinear and couples various fields. In order to solve the weak form, we use the finite element method in space and the finite difference method in time for the discretization of the computational domain. Numerical problems are circumvented by selecting the field equations carefully, and the weak form is assembled using standard shape functions. In order to examine the accuracy of the method, for the case of a linear elastic material under small deformations, we present and use an analytic solution. Comparison of the computation to the closed-form solution shows that the computational approach is reliable and models the jump of the electromagnetic fields across the interface between two different materials.     

Nonlinear Hyperbolic Systems: Nondegenerate Flux,Inner Speed Variation,and Graph Solutions

We study the Cauchy problem for general nonlinear strictly hyperbolic systems of partial differential equations in one space variable. First, we re-visit the construction of the solution to the Riemann problem and introduce the notion of a nondegenerate (ND) system. This is the optimal condition guaranteeing, as we show it, that the Riemann problem can be solved with finitely many waves only; we establish that the ND condition is generic in the sense of Baire (for the Whitney topology), so that any system can be approached by a ND system. Second, we introduce the concept of inner speed variation and we derive new interaction estimates on wave speeds. Third, we design a wave front tracking scheme and establish its strong convergence to the entropy solution of the Cauchy problem; this provides a new existence proof as well as an approximation algorithm. As an application, we investigate the time regularity of the graph solutions (X,U) introduced by LeFloch, and propose a geometric version of our scheme; in turn, the spatial component X of a graph solution can be chosen to be continuous in both time and space, while its component U is continuous in space and has bounded variation in time.     

A high-order parallel Eulerian-Lagrangian algorithm for advection-diffusion problems on unstructured meshes

In this paper, we present a high-order discontinuous Galerkin Eulerian-Lagrangian method for the solution of advection-diffusion problems on staggered unstructured meshes in two and three space dimensions. The particle trajectories are tracked backward in time by means of a high-order representation of the velocity field and a linear mapping from the physical to a reference system, hence obtaining a very simple and efficient strategy that permits to follow the Lagrangian trajectories throughout the computational domain. The use of an Eulerian-Lagrangian discretization increases the overall computational efficiency of the scheme because it is the only explicit method for the discretization of convective terms that admits large time steps without imposing a Courant-Friedrichs-Lewy–type stability condition. This property is fully exploited in this work by relying on a semi-implicit discretization of the incompressible Navier-Stokes equations, in which the pressure is discretized implicitly; thus, the sound speed does not play any role in the restriction of the maximum admissible time step. The resulting mild Courant-Friedrichs-Lewy stability condition, which is based only on the fluid velocity, is here overcome by the adoption of the Eulerian-Lagrangian method for the advection terms and an implicit scheme for the diffusive part of the governing equations. As a consequence, the novel algorithm is able to run simulation with a time step that is defined by the user, depending on the desired efficiency and time scale of the physical phenomena under consideration. Finally, a complete Message Passing Interface parallelization of the code is presented, showing that our approach can reach up to 96% of scaling efficiency.     

Numerical solutions of systems with (p,δ)‐structure using local discontinuous Galerkin finite element methods

In this paper, we present LDG methods for systems with (p,δ)‐structure. The unknown gradient and the nonlinear diffusivity function are introduced as auxiliary variables and the original (p,δ) system is decomposed into a first‐order system. Every equation of the produced first‐order system is discretized in the discontinuous Galerkin framework, where two different nonlinear viscous numerical fluxes are implemented. An a priori bound for a simplified problem is derived. The ODE system resulting from the LDG discretization is solved by diagonal implicit Runge–Kutta methods. The nonlinear system of algebraic equations with unknowns the intermediate solutions of the Runge–Kutta cycle is solved using Newton and Picard iterative methodology. The performance of the two nonlinear solvers is compared with simple test problems. Numerical tests concerning problems with exact solutions are performed in order to validate the theoretical spatial accuracy of the proposed method. Further, more realistic numerical examples are solved in domains with non‐smooth boundary to test the efficiency of the method. Copyright © 2014 John Wiley & Sons, Ltd.     

An efficient and stable finite element solver of higher order in space and time for nonstationary incompressible flow

In this paper, we present fully implicit continuous Galerkin–Petrov (cGP) and discontinuous Galerkin (dG) time‐stepping schemes for incompressible flow problems which are, in contrast to standard approaches like for instance the Crank–Nicolson scheme, of higher order in time. In particular, we analyze numerically the higher order dG(1) and cGP(2) methods, which are super convergent of third, resp., fourth order in time, whereas for the space discretization, the well‐known LBB‐stable finite element pair of third‐order accuracy is used. The discretized systems of nonlinear equations are treated by using the Newton method, and the associated linear subproblems are solved by means of a monolithic (geometrical) multigrid method with a blockwise Vanka‐like smoother treating all components simultaneously. We perform nonstationary simulations (in 2D) for two benchmarking configurations to analyze the temporal accuracy and efficiency of the presented time discretization schemes w.r.t. CPU and numerical costs. As a first test problem, we consider a classical ‘flow around cylinder’ benchmark. Here, we concentrate on the nonstationary behavior of the flow patterns with periodic oscillations and examine the ability of the different time discretization schemes to capture the dynamics of the flow. As a second test case, we consider the nonstationary ‘flow through a Venturi pipe’. The objective of this simulation is to control the instantaneous and mean flux through this device. Copyright © 2013 John Wiley & Sons, Ltd.     

Adaptive finite elements with large aspect ratio for mass transport in electroosmosis and pressure‐driven microflows

A space–time adaptive method is presented for the numerical simulation of mass transport in electroosmotic and pressure‐driven microflows in two space dimensions. The method uses finite elements with large aspect ratio, which allows the electroosmotic flow and the mass transport to be solved accurately despite the presence of strong boundary layers. The unknowns are the external electric potential, the electrical double layer potential, the velocity field and the sample concentration. Continuous piecewise linear stabilized finite elements with large aspect ratio and the Crank–Nicolson scheme are used for the space and time discretization of the concentration equation. Numerical results are presented showing the efficiency of this approach, first in a straight channel, then in crossing and multiple T‐form configuration channels. Copyright © 2009 John Wiley & Sons, Ltd.     

3D staggered Lagrangian hydrodynamics scheme with cell‐centered Riemann solver‐based artificial viscosity

The aim of the present work is the 3D extension of a general formalism to derive a staggered discretization for Lagrangian hydrodynamics on unstructured grids. The classical compatible discretization is used; namely, momentum equation is discretized using the fundamental concept of subcell forces. Specific internal energy equation is obtained using total energy conservation. The subcell force is derived by invoking the Galilean invariance and thermodynamic consistency. A general form of the subcell force is provided so that a cell entropy inequality is satisfied. The subcell force consists of a classical pressure term plus a tensorial viscous contribution proportional to the difference between the node velocity and the cell‐centered velocity. This cell‐centered velocity is an extra degree of freedom solved with a cell‐centered approximate Riemann solver. The second law of thermodynamics is satisfied by construction of the local positive definite subcell tensor involved in the viscous term. A particular expression of this tensor is proposed. A more accurate extension of this discretization both in time and space is also provided using a piecewise linear reconstruction of the velocity field and a predictor‐corrector time discretization. Numerical tests are presented in order to assess the efficiency of this approach in 3D. Sanity checks show that the 3D extension of the 2D approach reproduces 1D and 2D results. Finally, 3D problems such as Sedov, Noh, and Saltzman are simulated. Copyright © 2012 John Wiley & Sons, Ltd.     

A 3D finite volume scheme for solving the updated Lagrangian form of hyperelasticity

The finite volume discretization of nonlinear elasticity equations seems to be a promising alternative to the traditional finite element discretization as mentioned by Lee et al. [Computers and Structures (2013)]. In this work, we propose to solve the elastic response of a solid material by using a cell‐centered finite volume Lagrangian scheme in the current configuration. The hyperelastic approach is chosen for representing elastic isotropic materials. In this way, the constitutive law is based on the principle of frame indifference and thermodynamic consistency, which are imposed by mean of the Coleman–Noll procedure. It results in defining the Cauchy stress tensor as the derivative of the free energy with respect to the left Cauchy–Green tensor. Moreover, the materials being isotropic, the free‐energy is function of the left Cauchy–Green tensor invariants, which enable the use of the neo‐Hookean model. The hyperelasticity system is discretized using the cell‐centered Lagrangian scheme from the work of Maire et al. [J. Comput. Phys. (2009)]. The 3D scheme is first order in space and time and is assessed against three test cases with both infinitesimal displacements and large deformations to show the good accordance between the numerical solutions and the analytic ones. Copyright © 2016 John Wiley & Sons, Ltd.     

A new stable space–time formulation for two‐dimensional and three‐dimensional incompressible viscous flow

A space–time finite element method for the incompressible Navier–Stokes equations in a bounded domain in ?^d (with d=2 or 3) is presented. The method is based on the time‐discontinuous Galerkin method with the use of simplex‐type meshes together with the requirement that the space–time finite element discretization for the velocity and the pressure satisfy the inf–sup stability condition of Brezzi and Babu?ka. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. The stability proof and numerical results for some two‐dimensional problems are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Computation of unsteady viscous incompressible flows in generalized non‐inertial co‐ordinate system using Godunov‐projection method and overlapping meshes

Time‐dependent incompressible Navier–Stokes equations are formulated in generalized non‐inertial co‐ordinate system and numerically solved by using a modified second‐order Godunov‐projection method on a system of overlapped body‐fitted structured grids. The projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The second‐order Godunov method is applied for numerically approximating the non‐linear convection terms in order to provide a robust discretization for simulating flows at high Reynolds number. In order to obtain the pressure field, the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain so that the moving‐boundary problem can be solved economically. Numerical results are then presented to demonstrate the performance of this projection method for a variety of unsteady two‐ and three‐dimensional flow problems formulated in the non‐inertial co‐ordinate systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Flow simulation on moving boundary‐fitted grids and application to fluid–structure interaction problems

We present a method for the parallel numerical simulation of transient three‐dimensional fluid–structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non‐overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time‐dependent domains. To this end, we present a technique to solve the incompressible Navier–Stokes equation in three‐dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time‐dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes equations. Here the grid velocity is treated in such a way that the so‐called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well‐known MAC‐method to a staggered mesh in moving boundary‐fitted coordinates which uses grid‐dependent velocity components as the primary variables. To validate our method, we present some numerical results which show that second‐order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluid–structure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid–structure interaction problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Robust high-order semi-implicit backward difference formulas for Navier-Stokes equations

Taylor-Hood finite elements provide a robust numerical discretization of Navier-Stokes equations (NSEs) with arbitrary high order of accuracy in space. To match the accuracy of the lowest degree Taylor-Hood element, we propose a very efficient time-stepping methods for unsteady flows, which are based on high-order semi-implicit backward difference formulas (SBDF) and the inclusion of grad -div term in the NSE. To mitigate the impact on the numerical accuracy (in time) of the extrapolation of the nonlinear term in SBDF, several variants of nonlinear extrapolation formulas are investigated. The first approach is based on an extrapolation of the nonlinear advection term itself. The second formula uses the extrapolation of the velocity prior to the evaluation of the nonlinear advection term as a whole. The third variant is constructed such that it produces similar error on both velocity and pressure to that with fully implicit backward difference formulas methods at a given order of accuracy. This can be achieved by fixing one-order higher than usually done in the extrapolation formula for the nonlinear advection term, while keeping the same extrapolation formula for the time derivative. The resulting truncation errors (in time) between these formulas are identified using Taylor expansions. These truncation error formulas are shown to properly represent the error seen in numerical tests using a 2D manufactured solution. Lastly, we show the robustness of the proposed semi-implicit methods by solving test cases with high Reynolds numbers using one of the nonlinear extrapolation formulas, namely, the 2D flow past circular cylinder at Re=300 and Re = 1000 and the 2D lid-driven cavity at Re = 50 000 and Re = 100 000. Our numerical solutions are found to be in a good agreement with those obtained in the literature, both qualitatively and quantitatively.     

Three‐dimensional transient Navier–Stokes solvers in cylindrical coordinate system based on a spectral collocation method using explicit treatment of the pressure

A spectral collocation method is developed for solving the three‐dimensional transient Navier–Stokes equations in cylindrical coordinate system. The Chebyshev–Fourier spectral collocation method is used for spatial approximation. A second‐order semi‐implicit scheme with explicit treatment of the pressure and implicit treatment of the viscous term is used for the time discretization. The pressure Poisson equation enforces the incompressibility constraint for the velocity field, and the pressure is solved through the pressure Poisson equation with a Neumann boundary condition. We demonstrate by numerical results that this scheme is stable under the standard Courant–Friedrichs–Lewy (CFL) condition, and is second‐order accurate in time for the velocity, pressure, and divergence. Further, we develop three accurate, stable, and efficient solvers based on this algorithm by selecting different collocation points in r‐, ? ‐, and z‐directions. Additionally, we compare two sets of collocation points used to avoid the axis, and the numerical results indicate that using the Chebyshev Gauss–Radau points in radial direction to avoid the axis is more practical for solving our problem, and its main advantage is to save the CPU time compared with using the Chebyshev Gauss–Lobatto points in radial direction to avoid the axis. Copyright © 2010 John Wiley & Sons, Ltd.     

On unconditionally positive implicit time integration for the DG scheme applied to shallow water flows

We present a new unconditionally positivity‐preserving (PP) implicit time integration method for the DG scheme applied to shallow water flows. This novel time discretization enhances the currently used PP DG schemes, because in the majority of previous work, explicit time stepping is implemented to deal with wetting and drying. However, for explicit time integration, linear stability requires very small time steps. Especially for locally refined grids, the stiff system resulting from space discretization makes implicit or partially implicit time stepping absolutely necessary. As implicit schemes require a lot of computational time solving large systems of nonlinear equations, a much larger time step is necessary to beat explicit time stepping in terms of CPU time. Unfortunately, the current PP implicit schemes are subject to time step restrictions due to a so‐called strong stability preserving constraint. In this work, we hence give a novel approach to positivity preservation including its theoretical background. The new technique is based on the so‐called Patankar trick and guarantees non‐negativity of the water height for any time step size while still preserving conservativity. In the DG context, we prove consistency of the discretization as well as a truncation error of the third order away from the wet–dry transition. Because of the proposed modification, the implicit scheme can take full advantage of larger time steps and is able to beat explicit time stepping in terms of CPU time. The performance and accuracy of this new method are studied for several classical test cases. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence of Solutions for a Mathematical Model Related to Solid–Solid Phase Transitions in Shape Memory Alloys

We consider a strongly nonlinear PDE system describing solid–solid phase transitions in shape memory alloys. The system accounts for the evolution of an order parameter χ (related to different symmetries of the crystal lattice in the phase configurations), of the stress (and the displacement u), and of the absolute temperature ?. The resulting equations present several technical difficulties to be tackled; in particular, we emphasize the presence of nonlinear coupling terms, higher order dissipative contributions, possibly multivalued operators. As for the evolution of temperature, a highly nonlinear parabolic equation has to be solved for a right hand side that is controlled only in L ¹. We prove the existence of a solution for a regularized version by use of a time discretization technique. Then, we perform suitable a priori estimates which allow us pass to the limit and find a weak global-in-time solution to the system.