Semi-implicit Time Discretization Research Articles

Inflow-implicit/outflow-explicit finite volume methods for solving advection equations

We introduce a new class of methods for solving non-stationary advection equations. The new methods are based on finite volume space discretizations and a semi-implicit discretization in time. Its basic idea is that outflow from a cell is treated explicitly while inflow is treated implicitly. This is natural, since we know what is outflowing from a cell at the old time step but we leave the method to resolve a system of equations determined by the inflows to a cell to obtain the solution values at the new time step. The matrix of the system in our inflow-implicit/outflow-explicit (IIOE) method is determined by the inflow fluxes which results in an M-matrix yielding favorable stability properties for the scheme. Since the explicit (outflow) part is not always dominated by the implicit (inflow) part and thus some oscillations can occur, we build a stabilization based on the upstream weighted averages with coefficients determined by the flux-corrected transport approach [2,19] yielding high resolution versions, S1IIOE and S2IIOE, of the basic scheme. We prove that our new method is exact for any choice of a discrete time step on uniform rectangular grids in the case of constant velocity transport of quadratic functions in any dimension. We also show its formal second order accuracy in space and time for 1D advection problems with variable velocity. Although designed for non-divergence free velocity fields, we show that the basic IIOE scheme is locally mass conservative in case of divergence free velocity. Finally, we show L2-stability for divergence free velocity in 1D on periodic domains independent of the choice of the time step, and L∞-stability for the stabilized high resolution variant of the scheme. Numerical comparisons with the purely explicit schemes like the fully explicit up-wind and the Lax–Wendroff schemes were discussed in [13] and [14] where the basic IIOE was originally introduced. There it has been shown that the new scheme has good properties with respect to a balance of precision and CPU time related to a possible choice of larger time steps in our scheme. In this contribution we compare the new scheme and its stabilized variants with widely used fully implicit up-wind method. In this comparison our new schemes show better behavior with respect to stability and precision of computations for time steps several times exceeding the CFL stability condition. Our new stabilized methods are L∞ stable, second order accurate for any smooth solution and with accuracy of order 2/3 for solutions with moving discontinuities. This is opposite to implicit up-wind schemes which have accuracy order 1/2 only. All these properties hold for any choice of time step thus making our new method attractive for practical applications.

Optimal error estimate for semi-implicit space-time discretization for the equations describing incompressible generalized Newtonian fluids

In this paper we study the numerical error arising in the space-time approxima- tion of unsteady generalized Newtonian fluids which possess a stress-tensor with (p.�)-structure. A semi-implicit time-discretization scheme coupled with conforming inf-sup stable finite element space discretization is analyzed. The main result, which improves previous suboptimal estimates as those in (A. Prohl, and M. Ru uyziycka, SIAM J. Numer. Anal., 39 (2001), pp. 214-249) is the optimal O(k+h) error-estimate valid in the range p 2 (3/2,2), where k and h are the time-step and the mesh-size, respectively. Our results hold in three-dimensional domains (with periodic boundary conditions) and are uniform with respect to the degeneracy parameter � 2 (0,�0) of the extra stress tensor.

A Conservative Local Discontinuous Galerkin Method for the Schrödinger-KdV System

AbstractIn this paper we develop a conservative local discontinuous Galerkin (LDG) method for the Schrödinger-Korteweg-de Vries (Sch-KdV) system, which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Conservative quantities in the discrete version of the number of plasmons, energy of the oscillations and the number of particles are proved for the LDG scheme of the Sch-KdV system. Semi-implicit time discretization is adopted to relax the time step constraint from the high order spatial derivatives. Numerical results for accuracy tests of stationary traveling soliton, and the collision of solitons are shown. Numerical experiments illustrate the accuracy and capability of the method.

An all-at-once approach for the optimal control of the unsteady Burgers equation

We apply an all-at-once method for the optimal control of the unsteady Burgers equation. The nonlinear Burgers equation is discretized in time using the semi-implicit discretization and the resulting first order optimality conditions are solved iteratively by Newton’s method. The discretize then optimize approach is used, because it leads to a symmetric indefinite saddle point problem. Numerical results for the distributed unconstrained and control constrained problems illustrate the performance of the all-at-once approach with semi-implicit time discretization.

Adhesive contact delaminating at mixed mode, its thermodynamics and analysis

An adhesive unilateral contact problem between visco-elastic heat-conductive 1 bodies in linear Kelvin-Voigt rheology is scrutinized. The flow-rule for debonding the adhesive is considered rate-independent, unidirectional, and non-associative due to dependence on the mixity of modes of delamination, namely of Mode I (opening) and of Mode II (shearing). Such mode-mixity dependence of delamination is a very pronounced (and experimentally confirmed) phenomenon typically considered in engineering models. An anisothermal, thermodynamically consistent model is derived, considering a heat-conductive viscoelastic material and the coupling via thermal expansion and adhesion-depending heat transition through the contact surface. We prove the existence of weak solutions by passing to the limit in a carefully designed semi-implicit time-discretization scheme.

A Model of Rupturing Lithospheric Faults with Reoccurring Earthquakes

An isothermal small-strain model based on the concept of generalized standard materials is devised, combining Maxwell-type rheology, damage, and perfect plasticity in the bulk. An interface analogue of the model is prescribed at the lithospheric faults, exploiting concepts of adhesive contacts with interfacial plasticity. The model covers simultaneously features such as rupturing of the fault zone accompanied with weakening/healing effects and also seismic waves emission and propagation connected with the sudden ruptures of the fault or a fluid-like aseismic response between the ruptures. A stable numerical strategy based on semi-implicit discretization in time is devised, and its convergence is shown. Numerical simulations documenting the capacity of the model to simulate earthquakes with repeating occurrence are performed, too.

Rates of Convergence for Discretizations of the Stochastic Incompressible Navier--Stokes Equations

We show strong convergence with rates for an implicit time discretization, a semi-implicit time discretization, and a related finite element based space-time discretization of the incompressible Navier--Stokes equations with multiplicative noise in two space dimensions. We use higher moments of computed iterates to optimally bound the error on a subset $\Omega_\kappa$ of the sample space $\Omega$, where corresponding paths are bounded in a proper function space, and $\mathbb{P}[\Omega_\kappa] \to 1$ holds for vanishing discretization parameters. This implies convergence in probability with rates, and motivates a practicable acception/rejection criterion to overcome possible pathwise explosion behavior caused by the nonlinearity. It turns out that it is the interaction of Lagrange multipliers with the stochastic forcing in the scheme which limits the accuracy of general discretely LBB-stable space discretizations, and strategies to overcome this problem are proposed.

On the Approximation of the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model I: A New Weighted Formulation and an Optimal Spectral-Galerkin Algorithm in Two Dimensions

We propose a new weighted weak formulation for the Fokker–Planck equation of the finitely extensible nonlinear elastic dumbbell model and prove its well-posedness in weighted Sobolev spaces. We also propose simple and efficient semi-implicit time-discretization schemes which are unconditionally stable, i.e., the step size of time marching does not depend on the number of the bases used in configurational space. We then restrict ourselves to the two-dimensional case and construct two Fourier–Jacobi spectral-Galerkin algorithms which enjoy the following properties: (i) they are unconditionally stable, spectrally accurate, and of optimal computational complexity; (ii) they conserve the volume and provide accurate approximation to higher-order moments of the distribution function; and (iii) they can be easily extended to coupled nonhomogeneous systems. Numerical results are presented to show how to choose a proper weight to get the best numerical results of the distribution function and the polymer stress.

Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis

We are concerned with the finite-element approximation for the Keller-Segel system that describes the aggregation of slime molds resulting from their chemotactic features. The scheme makes use of a semi-implicit time discretization with a time-increment control and Baba-Tabata's conservative upwind finite-element approximation in order to realize the positivity and mass conservation properties. The main aim is to present error analysis that is an application of the discrete version of the analytical semigroup theory.

Primary charge effects on prolate spheroids with moderate aspect ratios

In this article, we investigate the behavior of charged spheroidal colloids with moderate aspect ratios in linear flow fields. We use direct numerical simulation with body-fitted grids for the solution of the Stokes–Poisson–Nernst–Planck system to include all non-linear effects. Therefore, we propose an efficient semi-implicit time discretization based on a splitting of the Stokes equation. We will study the effects of the electric double layer on the forces, torques and on the motion of spheroidal particles. For low Reynolds numbers, we find approximating linear expressions between the ambient fluid flow and the force and torque on the particle. The description of this linear behavior is based on the resistance functions, whose dependencies on the Debye length and the zeta potential are investigated. It is recovered that the resistance functions obey a quadratic dependence on the zeta potential in the small zeta potential regime. For low values of the zeta potential, approximate formulas for the resistance functions are given. The approximation properties are carefully studied by comparing the approximate results with direct numerical simulations. For the case of a shear flow, the approximate formulas can be used to avoid time-consuming direct numerical simulations.

All Speed Scheme for the Low Mach Number Limit of the Isentropic Euler Equations

AbstractAn all speed scheme for the Isentropic Euler equations is presented in this paper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficulty in the low Mach regime. The key idea of our all speed scheme is the special semi-implicit time discretization, in which the low Mach number stiff term is divided into two parts, one being treated explicitly and the other one implicitly. Moreover, the flux of the density equation is also treated implicitly and an elliptic type equation is derived to obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared with previous semi-implicit methods, firstly, nonphysical oscillations can be suppressed by choosing proper parameter, besides, only a linear elliptic equation needs to be solved implicitly which reduces much computational cost. We develop this semi-implicit time discretization in the framework of a first order Local Lax-Friedrichs (or Rusanov) scheme and numerical tests are displayed to demonstrate its performances.

Efficient solution strategy for the semi-implicit discontinuous Galerkin discretization of the Navier–Stokes equations

We deal with the numerical solution of the system of the compressible Navier–Stokes equations with the aid of the interior penalty Galerkin method. We employ a semi-implicit time discretization which leads to the solution of a sequence of linear algebraic systems. We develop an efficient strategy for the solution of these systems. It is based on a simple adaptive technique for the choice of the time step and a relatively weak stopping criterion for iterative linear algebraic solvers. The presented numerical experiments show that the proposed strategy is efficient for steady-state problems using various grids, polynomial degrees of approximations and flow regimes. Finally, we apply this strategy with a minor modification to an unsteady flow.

Modeling and computation of two phase geometric biomembranes using surface finite elements

Biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions. The modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces. This leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints. In this paper we propose a new computational tool for computing equilibria based on an L 2 relaxation flow for the total energy in which the line energy is approximated by a surface Ginzburg–Landau phase field functional. The relaxation dynamics couple a nonlinear fourth order geometric evolution equation of Willmore flow type for the membrane with a surface Allen–Cahn equation describing the lateral decomposition. A novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface, the mean curvature vector and the surface phase field function. The resulting variational formulation uses H 1 spaces, and we employ triangulated surfaces and H 1 conforming quadratic surface finite elements for approximating solutions. Together with a semi-implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only. Numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells, discocytes and starfishes with lateral phase separation.

Three-dimensional, fully adaptive simulations of phase-field fluid models

We present an efficient numerical methodology for the 3D computation of incompressible multi-phase flows described by conservative phase-field models. We focus here on the case of density matched fluids with different viscosity (Model H). The numerical method employs adaptive mesh refinements (AMR) in concert with an efficient semi-implicit time discretization strategy and a linear, multi-level multigrid to relax high order stability constraints and to capture the flow’s disparate scales at optimal cost. Only five linear solvers are needed per time-step. Moreover, all the adaptive methodology is constructed from scratch to allow a systematic investigation of the key aspects of AMR in a conservative, phase-field setting. We validate the method and demonstrate its capabilities and efficacy with important examples of drop deformation, Kelvin–Helmholtz instability, and flow-induced drop coalescence.

Mixed finite element method for electrowetting on dielectric with contact line pinning

We present a mixed finite element method for a model of the flow in a Hele-Shaw cell of 2-D fluid droplets surrounded by air driven by surface tension and actuated by an electric field. The application of interest regards a micro-fluidic device called ElectroWetting on Dielectric (EWOD). Our analysis first focuses on the time discrete (continuous in space) problem and is presented in a mixed variational framework, which incorporates curvature as a natural boundary condition. The model includes a viscous damping term for interface motion, as well as contact line pinning (sticking of the interface) and is captured in our formulation by a variational inequality. The semi-discrete problem uses a semiimplicit time discretization of curvature. We prove the well-posedness of the semi-discrete problem and fully discrete problem when discretized with iso-parametric finite elements. We derive a priori error estimates for the space discretization. We also prove the convergence of an Uzawa algorithm for solving the semi-discrete EWOD system with inequality constraint. We conclude with a discussion about experimental orders of convergence.

A New Level Set Method for Motion in Normal Direction Based on a Semi-Implicit Forward-Backward Diffusion Approach

We introduce a new level set method for motion in normal direction. It is based on a formulation in the form of a second order forward-backward diffusion equation. The equation is discretized by the finite volume method. We propose a semi-implicit time discretization taking into account the forward diffusion part of the solution in an implicit way, while the backward diffusion part is treated explicitly. When forward diffusion dominates, a straightforward reconstruction of the solution is used, while larger (smoothing) stencils are used when backward diffusion dominates. The method is precise on coarse grids and is second order accurate for smooth solutions. Numerical experiments show an optimal coupling of time and space steps with $\tau=h$, and no stronger CFL condition is required. Numerical tests with the scheme are discussed on representative examples.

An optimization-based algorithm for Coulomb's frictional contact

The main goal of this paper is to propose a stable algorithm to compute friction forces governed by Coulomb's law in the course of the simulation of a nonsmooth Lagrangian dynamical system. The problem appears in computational mechanics, to simulate the dynamics of granular materials, robots, etc . Using a classical impulse-velocity formulation of Coulomb's law to model friction, and a semi-implicit time discretization scheme, we get a set of linear, non-linear and complementarity equations which has to be solved at each timestep. Two mutually dual parametric convex optimization problems coupled with a fixed point equation appear naturally. By solving (one of) these optimization problems iteratively within a damped nonsmooth-Newton algorithm, it is possible to decrease some infeasibility criterion and hopefully converge to a solution of the system. Numerical results are provided, which show that the number of iterations needed by the algorithm is very small in general and that the method is stable.

Optimal Error Estimates for a Semi-Implicit Euler Scheme for Incompressible Fluids with Shear Dependent Viscosities

Certain rheological behaviors of fluids in engineering sciences are modeled by power law ansatz with $p\in(1,2]$. In the present paper a semi-implicit time discretization scheme for such fluids is proposed. The main result is the optimal $\mathcal{O}(k)$ error estimate, where k is the time step size. Our results hold in the range $p\in(3/2,2]$ (in the three-dimensional setting) for strong solutions of the continuous problem, whose existence is guaranteed under appropriate assumptions on the data. The estimates are uniform with respect to the degeneracy parameter $\delta\in[0,\delta_0]$ of the extra stress tensor. Additional regularity properties of the solution of the discrete problem are proved.

Nucleation and growth by a phase field crystal (PFC) model

We review the derivation of a phase field crystal (PFC) model from classical density functional theory (DFT). Through a gradient flow of the Helmholtz free energy functional and appropriate approximations of the correlation functions, higher order nonlinear equations are derived for the evolution of a time averaged density. The equation is solved by finite elements using a semi-implicit time discretization.

Adaptive solution techniques for simulating underwater explosions and implosions

Adaptive solution techniques are presented for simulating underwater explosions and implosions. The liquid is assumed to be an adiabatic fluid and the solution in the gas is assumed to be uniform in space. The solution in water is integrated in time using a semi-implicit time discretization of the adiabatic Euler equations. Results are presented either using a non-conservative semi-implicit algorithm or a conservative semi-implicit algorithm. A semi-implicit algorithm allows one to compute with relatively large time steps compared to an explicit method. The interface solver is based on the coupled level set and volume-of-fluid method (CLSVOF) [M. Sussman, A second order coupled level set and volume-of-fluid method for computing growth and collapse of vapor bubbles, J. Comput. Phys. 187 (2003) 110–136; M. Sussman, E.G. Puckett, A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows, J. Comput. Phys. 162 (2000) 301–337]. Several underwater explosion and implosion test cases are presented to show the performances of our proposed techniques.