Global asymptotic stability of one time-continuous model and its time-discrete variant for basic virus dynamics

This paper describes a flexible time-stepping scheme for a recently developed hybrid field-circuit solver based on the extended time-domain finite element method (TDFEM) to alleviate the limitation on the use of a system-wide global time-step size. The proposed time-stepping scheme generalizes the strict synchronous coupling mechanism between the FEM and circuit subsystems and allows the signals in the different subsystems to be tracked and sampled at different time-step sizes. The signals from a slow subsystem with a larger time-step size are extrapolated, when necessary, for updating the signals in a fast subsystem with a smaller time-step size. The capability of the hybrid field-circuit solver with the proposed time-stepping scheme is further enhanced by the application of a tree-cotree splitting technique to the FEM subsystem, which helps reduce the iteration count per time step for a preconditioned iterative solution when the time-step size of the FEM subsystem becomes relatively large. With the flexibility of choosing subsystem-specific time-step sizes, the proposed time-stepping scheme improves the computational efficiency of the existing TDFEM-based hybrid field-circuit solver especially when the computational cost associated with the slow subsystems is much higher than that associated with the fast subsystems.

Numerical simulations of binary black holes---an important predictive tool for the detection of gravitational waves---are computationally expensive, especially for binaries with high mass ratios or with rapidly spinning constituent holes. Existing codes for evolving binary black holes rely on explicit timestepping methods, for which the timestep size is limited by the smallest spatial scale through the Courant-Friedrichs-Lewy condition. Binary inspiral typically involves spatial scales (the spatial resolution required by a small or rapidly spinning hole) which are orders of magnitude smaller than the relevant (orbital, precession, and radiation-reaction) timescales characterizing the inspiral. Therefore, in explicit evolutions of binary black holes, the timestep size is typically orders of magnitude smaller than the relevant physical timescales. Implicit timestepping methods allow for larger timesteps, and they often reduce the total computational cost (without significant loss of accuracy) for problems dominated by spatial rather than temporal error, such as for binary-black-hole inspiral in corotating coordinates. However, fully implicit methods can be difficult to implement for nonlinear evolution systems like the Einstein equations. Therefore, in this paper we explore implicit-explicit (IMEX) methods and use them for the first time to evolve black-hole spacetimes. Specifically, as a first step toward IMEX evolution of a full binary-black-hole spacetime, we develop an IMEX algorithm for the generalized harmonic formulation of the Einstein equations and use this algorithm to evolve stationary and perturbed single-black-hole spacetimes. Numerical experiments explore the stability and computational efficiency of our method.

Conservative explicit local time-stepping schemes for the shallow water equations

This paper describes an explicit transient analysis method stabilized for an arbitrary time step size for fast simulation of a multilayered power delivery network (PDN). A time step size used in an existing block-type leapfrog scheme is forced to be small if there exists small reactances in a circuit. Such small reactances are extracted from the small meshes, which are used to represent small apertures on the conductor planes. Therefore, the conditionally-stable explicit method is not suitable for the circuit including the small reactances. The stabilized explicit method (SEM) can remove instability related to the low reactances parts and enables to use a relatively-large time step size compared with the explicit leapfrog scheme. SEM is applied to transient simulations of an example multilayered PDN with a number of circular apertures to evaluate its adequacy. Numerical results show that our approach can perform much faster simulation than the block-type leapfrog scheme and a conventional SPICE-like simulator.

The constantly increasing demand for efficient and precise computational solvers becomes the crucial factor deciding about usability of a given domain specific simulation software. The main idea of this article is the use of eigenvalues of amplification matrices to determine the size of time step in modeling of solidification. As far as numerical simulations are concerned it is very important to obtain solutions which are stable and physically correct. It is acquired by fulfilling many assumptions and conditions during the construction a numerical model and carrying out computer simulations. One of the conditions is a proper selection of time step. The size of time step has a great impact on the stability of used time integration schemes (e.g. explicit scheme), or on a proper image of physical phenomena occurring during the simulation (e.g. implicit scheme). The eigenvalues of amplification matrix in governing equations influence on the appropriate selection of size of time step in computer simulations. Hence, it allows to better fit the size of time step and time integration scheme for modeled structure.

The time domain analysis of eddy current problems often requires the simulation of long time intervals, e.g. until a steady state is reached. Fast-switching excitations e.g. in pulsedwidth modulated signals require in addition very small time step sizes that significantly increase computation time. To speed up the simulation, parallel-in-time methods can be used. In this paper, we investigate the combination of explicit and implicit time integration methods in the context of the parallel-in-time method Parareal and using a simplified model for the coarse propagator.

Reservoir simulations for complex multiphase flow and transport problems often suffer from non-linear solver convergence issues. These manifest in the form of restrictively small time-step sizes even while using unconditionally stable fully implicit schemes. These problems are further compounded when a local mesh refinement is used to accurately represent reservoir parameters available such as permeability, porosity, etc., at multiple spatial scales. We discuss a domain decomposition approach that allows different time-step sizes and mesh refinements in different subdomains (Singh and Wheeler (2018)) of the reservoir that circumvents these issues without compromising computational efficiency and prediction accuracy. This approach extends the well-known methodology of local mesh refinement in space (Wheeler et al. (2002)) to time. Our numerical experiments indicate that non-linear solvers fail to converge, to the desired tolerance, due to large non-linear residuals in a smaller subdomain. We exploit this feature to identify subdomains where smaller time-step sizes are necessary while using large time-step sizes in the rest of the reservoir domain. The three key components of our approach are: (1) a space-time, enhanced velocity, domain decomposition approach that allows different mesh refinements and time-step sizes in different subdomains while preserving local mass conservation, (2) a residual based error estimator to identify or mark regions (or subdomains) that pose non-linear convergence issues, and (3) a fully coupled monolithic solver is also presented that solves the coarse and fine subdomain problems, both in space and time, simultaneously. This solution scheme is fully implicit and is therefore unconditionally stable. The results indicate that using large time-step sizes for the entire reservoir domain poses serious non-linear solver convergence issues. Although using a smaller time step size for the entire domain reduces the convergence issues, it also results in substantial computational overheads. The proposed space-time domain decomposition approach, with smaller time-step sizes in a subdomain and large time-step sizes everywhere else, circumvents the non-linear convergence issue without adding computational costs. Additionally, a space-time monolithic solver renders a massively parallel, time concurrent framework for solving flow and transport problems in subsurface porous media. Since the proposed approach is similar to the widely used finite difference scheme, it can be easily integrated in any existing legacy reservoir simulator.

Non-iterative adaptive time-stepping scheme with temporal truncation error control for simulating variable-density flow

Partially explicit splitting method for a multi-physics problem

ABSTRACTIn this article, we introduce a fully implicit, linearly extrapolated second-order backward difference time-stepping scheme for solving a time dependent non-homogeneous magnetohydrodynamic system for electrically conducting fluids. The extrapolated time-stepping scheme is used for time discretization and the mixed finite element method is used for spatial discretization. We first prove unconditional energetic stability without introducing an undesirable exponential Gronwall constant. Complete error analysis is provided without assuming any stability condition or restrictions on the time-step size. Numerical experiments are presented to confirm the theoretical convergence results and efficiency of the scheme.

An explicit stable finite difference method for the Allen–Cahn equation

Chapter 6 - Temporal Discretisation

In this paper, we consider an HIV infection model with virus-to-cell infection, cell-to-cell transmission, intracellular delay and mitosis of uninfected cells. The basic reproduction number is calculated by using the method of the next generation matrix. By comparison arguments, it is proved that when the basic reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, the existence of Hopf bifurcation and stability switch at the chronic-infection equilibrium of the model with or without intracellular delay is established. Further, by constructing Lyapunov functionals, sufficient conditions are obtained for the global asymptotic stability of the chronic-infection equilibrium when the cell-to-cell transmission is negligible. Numerical simulations are carried out to illustrate the main theoretical results. The normal form is calculated to determine the bifurcation direction and stability, as well as amplitude and period of bifurcating periodic solutions when the intracellular delay is absent.

Stability analysis of within-host parasite models with delays

The problem of stability preservation under discretization of some classes of nonlinear differential equations systems is studied. Persidskii systems, Lurie systems of indirect control, and systems whose right-hand sides have a canonical structure are considered. It is assumed that the zero solutions of these systems are globally asymptotically stable. Conditions are determined that guarantee the asymptotic stability of the zero solutions for the corresponding difference systems. Previously, such conditions were established for the case where discretization was carried out using the explicit Euler method. In this paper, difference schemes are constructed on the basis of the implicit Euler method. For the obtained discrete systems, theorems on local and global asymptotic stability are proved, estimates of the time of transient processes are derived. For systems with a canonical structure of right-hand sides, based on the approach of V. I. Zubov, a modified implicit computational scheme is proposed that ensures the matching of the convergence rate of solutions to the origin for the differential and corresponding difference systems. It is shown that implicit computational schemes can guarantee the preservation of asymptotic stability under less stringent constraints on the discretization step and right-hand sides of the systems under consideration compared to the constraints obtained using the explicit method. An example is presented illustrating the obtained theoretical conclusions.