Operator-splitting methods for the 2D convective Cahn–Hilliard equation

Fully implicit schemes with second‐order time evolutions have been applied to simulate nonlinear diffusion problems precisely for a long time, but there is seldom theoretical study for either their convergence properties or efficient iterations. Here, a second‐order time evolution fully implicit scheme for two‐dimensional nonlinear divergence diffusion problem is analyzed. The unique existence of its solution is given. Two new methods are provided to prove its convergence, including entire inductive hypothesis reasoning and a two‐step reasoning process. Rigorous analysis shows the scheme is stable; its solution has second‐order convergence in both space and time to the exact solution of the problem. The convergence is applied to analyze a Newton iteration accelerating the computation and show its quadratic convergent speed and second‐order accuracy. The reasoning techniques also adapt to first‐order time accuracy schemes, and can be extended to analyze a wide class of nonlinear schemes for nonlinear problems. Numerical tests highlight the theoretical results and demonstrate the high performance of the algorithms. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 121–140, 2016

A second order front tracking method is developed for solving the Euler equations of inviscid fluid dynamics numerically. Front tracking methods are usually limited to first order accuracy, since they are based on a piecewise constant approximation of the solution. Here the second order convergence is achieved by building a piecewise linear reconstruction of the piecewise constant front tracking solution in a post-processing step. The linearization is performed by decomposing the piecewise constant solution of the hyperbolic system into its wave components and by linearizing the wave solutions separately. In order to achieve a physically correct linearization, the front types of the previously developed improved front tracking method are employed. It is illustrated numerically for the one-dimensional unsteady interacting blast waves problem and a two-dimensional supersonic airfoil flow validation study that the proposed front tracking method can achieve second order convergence also in the presents of strong discontinuities.

This paper examines the health-conscious optimal control of lithium-ion batteries. The paper focuses specifically on the optimizing battery control policies to minimize solid electrolyte interphase (SEI) layer growth. However, its approach is applicable to other degradation mechanisms. The paper exploits the fact, established by the mathematical control literature1–4, that Fick’s second law of diffusion is differential flat for both linear and nonlinear diffusion problems. Differential flatness only applies to each diffusion medium in the battery separately. In the single particle model (SPM), for instance, each electrode’s diffusion sub-model alone is differential flat. Ensuring differential flatness for the entire battery is a two-step process. First, one must truncate redundant integrators from the two electrodes’ diffusion sub-models. Second, one must then relate the truncated integral state variable to the remaining state variables through an affine transformation. The differential flatness of the resulting battery model makes it possible to solve the health-conscious battery management problem efficiently using pseudospectral methods. This exploitation of the structure (i.e., differential flatness) of lithium-ion battery dynamics for more computationally efficient trajectory optimization is novel and significant. The existing literature already examines the problem of health-conscious optimal battery charging and discharging5,6. However, the optimal control tools used in this literature are not tailored to exploit the structure (e.g., differential flatness) of lithium-ion battery models. This can lead to higher computational burdens compared to the novel and more efficient approach we present here. The approach has the added advantage that it can be used for both offline trajectory optimization and online model-predictive control.The paper demonstrates the above ideas using the SPM. The optimization problem is formulated by equation [1-6] and the SPM with film growth model. The cost function in Equation [1] aims to track the reference state of charge (SOC) and at the same time minimize the film growth rate, which is defined by equation [7]-[14] in Ramadass’s paper7. The weight β represents a tradeoff/balance between aggressive charging and battery degradation. Constraints [2-6] place limits on battery charge/discharge current, reflecting battery management hardware capabilities. Furthermore, these constraints limit battery SOC to prevent over-charging and over-discharging. Finally, these constraints also place bounds on battery state variables contributing to damage phenomena such as lithium plating and mechanical degradation. For instance, we bound the concentration gradients in the battery cell and the overpotentials driving the lithium plating side reaction8–10. The optimization approach presented in this paper transforms these equations and constraints from a dynamic programming problem to a nonlinear programming (NLP) problem. Solving this NLP problem using traditional optimization methods leads to a health-conscious battery management policy. The value of this paper lies not in this policy, but rather in the computational efficiency with which it is obtained. Specifically, to the best of the authors’ knowledge, this paper represents the first attempt to exploit lithium-ion battery model structure for more efficient solution of the health-conscious optimal management problem.ACKNOWLEDGMENTSThe research was funded by ARPA-E AMPED program grant # 0675-1565. The authors gratefully acknowledge this support.REFERENCES1. M. FLIESS, J. LÉVINE, P. MARTIN, and P. ROUCHON, Int. J. Control, 61, 1327–1361 (1995).2. B. Laroche, P. Martin, and P. Rouchon, Int. J. Robust Nonlinear Control, 10, 629–643 (2000).3. M. Fliess and R. Marquez, Int. J. Control, 73, 606–623 (2000).4. T. Meurer, Automatica, 47, 935–949 (2011).5. R. Klein and N. Chaturvedi, Am. Control … (2011).6. V. Boovaragavan and V. R. Subramanian, J. Power Sources, 173, 1006–1011 (2007).7. P. Ramadass, B. Haran, P. M. Gomadam, R. White, and B. N. Popov, J. Electrochem. Soc., 151, A196 (2004).8. J. Christensen and J. Newman, J. Solid State Electrochem., 10, 293–319 (2006).9. X. Zhang, A. M. Sastry, and W. Shyy, J. Electrochem. Soc., 155, A542 (2008).10. N. Chaturvedi, R. Klein, J. Christensen, J. Ahmed, and A. Kojic, IEEE Control Syst. Mag., 30, 49–68 (2010).

A new front tracking scheme for the ultra-relativistic Euler equations

A two-dimensional front tracking method based on conservation for scalar conservation law and for the Euler system of gas dynamics is being developed. In the method, discontinuities are tracked by enforcing the conservation properties of the PDEs. Unlike the traditional front tracking methods, for which the conservation is only a property that may or may not be preserved, in this front tracking method the conservation is the mechanism by which the tracking is performed. The method is also extended to treat the reflection wall boundary in the Euler system. As an attempt to treat the interactions of discontinuities by enforcing the conservation properties without solving two-dimensional Riemann problems, three special cases of treatment of discontinuity interactions in scalar conservation law and in the Euler system of gas dynamics are studied. Finally, numerical experiments are implemented in both scalar and system cases to show the efficiency of the method.

Front-Tracking Methods

A note on front tracking for the Keyfitz–Kranzer system

An improved Front-Tracking technique for the simulation of mass transfer in dense bubbly flows

Nonclassical Shocks and the Cauchy Problem for Nonconvex Conservation Laws

In this chapter we consider the Cauchy problem for a scalar conservation law. Our goal is to show that, subject to certain conditions, there exists a unique solution to the general initial value problem. Our method will be completely constructive, and we shall exhibit a procedure by which this solution can be constructed. This procedure is, of course, front tracking. The basic ingredient in the front-tracking algorithm is the solution of the Riemann problem.

A new 3D third Order Front-Tracking method and its application to 2D flow with phase change is presented. The originality of this method lies in the meshing of the computational domain it generates: At each time step, the highly deformed interface is a grid surface. This allows an accurate computation of the physical field in the vicinity of the interface as an exact quantification of their eventual jumps at the interface. To step toward the understanding of the physical processes involved in liquid film evaporation, the 2D formulation of this new front tracking method is used by merging it in a numerical model solving the conservation equations in a quadratic finite element formulation. The heat transport equation is solved in the three different phases (solid, liquid and vapor) as the Navier-Stokes equations are solved in the two fluids. The temperature gradient discontinuity at the liquid vapor interface provides local value of the evaporative flux density that is directly linked to the interface velocity jump through mass conservation principle and used as boundary conditions for the two fluid flow computations. Testing on academic cases and application to axysimmetric film evaporation including comparison with experiments are shown.

The application of a front tracking method to a nonstrictly hyperbolic system of conservation laws is described in one space dimension. The front tracking method is based on approximate solutions of Riemann problems. The method is compared with the random choice scheme and the upwind scheme.

Simulating phase transformations by a meshless method with front tracking

We study numerical methods for time-dependent partial differential equations describing wave propagation, primarily applied to problems in quantum dynamics governed by the time-dependent Schrodinger equation (TDSE). We consider both methods for spatial approximation and for time stepping. In most settings, numerical solution of the TDSE is more challenging than solving a hyperbolic wave equation. This is mainly because the dispersion relation of the TDSE makes it very sensitive to dispersion error, and infers a stringent time step restriction for standard explicit time stepping schemes. The TDSE is also often posed in high dimensions, where standard methods are intractable.The sensitivity to dispersion error makes spectral methods advantageous for the TDSE. We use spectral or pseudospectral methods in all except one of the included papers. In Paper III we improve and analyse the accuracy of the Fourier pseudospectral method applied to a problem with limited regularity, and in Paper V we construct a matrix-free spectral method for problems with non-trivial boundary conditions. Due to its stiffness, the TDSE is most often solved using exponential time integration. In this thesis we use exponential operator splitting and Krylov subspace methods. We rigorously prove convergence for force-gradient operator splitting methods in Paper IV. One way of making high-dimensional problems computationally tractable is low-rank approximation. In Paper VI we prove that a splitting method for dynamical low-rank approximation is robust to singular values in the approximation approaching zero, a situation which is difficult to handle since it implies strong curvature of the approximation space.

This paper intends to provide a brief review of the current well-posedness theory for hyperbolic systems of conservation laws in one space dimension, also pointing out open problems and possible research directions.