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The microbeam system at Tohoku University has various applications. Recently higher spatial resolution down to several hundred nm and higher beam current with the resolution of several μm were required. To meet these requirements, a triplet lens system was installed. While the triplet system has higher demagnification, the chromatic aberration is much larger than in the doublet system. To achieve better performance in the triplet system, improvements in the energy resolution of the accelerator are required. Various sources of accelerator voltage ripples were investigated. The high voltage generating circuit was symmetrized and the noise components were reduced to minimize the voltage ripple. The voltage ripple of the accelerator for low-frequency components was reduced to around 70 V. The voltage ripple of the 120-kHz component was 140 V_p-p.

We consider a phase field model for molecular beam epitaxial growth with slope selection with the goal of determining linear energy stable time integration methods for the dynamics. Stable methods for this model have been found via a concave-convex splitting of the dynamics, but this approach generally leads to a nonlinear update equation. We seek a linear energy stable method to allow for simple and efficient time marching with fast Fourier transforms. Our approach is to parametrize a class of semi-implicit methods and perform unconditional von Neumann stability analysis to identify the region of stability in parameter space. Since unconditional von Neumann stability does not ensure energy stability, we perform extensive numerical tests and find strong agreement between the predicted and observed stable regions of parameter space. This analysis elucidates a novel feature that the stability region in parameter space differs for a mono-domain system (single equilibrium slope) versus a many-domain system (coarsening facets from an initially flat surface). The utility of these steps is then demonstrated by a comparison of the coarsening dynamics for isotropic and anisotropic variants of the model.

We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn–Hilliard–Navier–Stokes phase- field model, prove its energy stability, and carry out error analysis for the corresponding Cahn–Hilliard–Stokes model only. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish second-order error estimates both in time and space for phase-field variable, chemical potential, velocity and pressure in different discrete norms for the Cahn–Hilliard–Stokes phase-field model. We also provide numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of our scheme.

We construct first- and second-order time discretization schemes for the Cahn–Hilliard–Navier–Stokes system based on the multiple scalar auxiliary variables (MSAV) approach for gradient systems and (rotational) pressure-correction for Navier–Stokes equations. These schemes are linear, fully decoupled, unconditionally energy stable, and only require solving a sequence of elliptic equations with constant coefficients at each time step. We carry out a rigorous error analysis for the first-order scheme, establishing optimal convergence rate for all relevant functions in different norms. We also provide numerical experiments to verify our theoretical results.

In this paper, we study the stability, accuracy and convergence behavior of various numerical schemes for phase-field modeling through a simple ODE model. Both theoretical analysis and numerical experiments are carried out on this ODE model to demonstrate the limitation of most numerical schemes that have been used in practice. One main conclusion is that the first-order fully implicit scheme is the only robust algorithm for phase-field simulations while all other schemes (that have been analyzed) may have convergence issue if the time step size is not exceedingly small. More specifically, by rigorous analysis in most cases, we have the following conclusions:

• (i)The first-order fully implicit scheme converges to the correct steady state solution for all time step sizes. In the case of multiple solutions, one of the solution branches always converges to the correct steady state solution.
• (ii)The first-order convex splitting scheme, which is equivalent to the first-order fully implicit scheme with a different time scaling, always converges to the correct steady state solution but may seriously lack numerical accuracy for transient solutions.
• (iii)For the second-order fully implicit and convex splitting schemes, for any time step size $\delta t>0$, there exists an initial condition $u_0$, with $\left|u_0\right|>1$, such that the numerical solution converges to the wrong steady state solution.
• (iv)For $\left|u_0\right|\le 1$, all second-order schemes studied in this paper converge to the correct steady state solution although severe numerical oscillations occur for most of them if the time step size is not sufficiently small.
• (v)An unconditionally energy-stable scheme (such as the modified Crank–Nicolson scheme) is not necessarily better than a conditionally energy-stable scheme (such as the Crank–Nicolson scheme).

Most, if not all, of the above conclusions are expected to be true for more general Allen–Cahn and other phase-field models.

In this paper, for the Allen–Cahn equation, we obtain the error estimate of the temporal semi-discrete scheme, and the fully-discrete finite element numerical scheme, both of which are based on the invariant energy quadratization (IEQ) time-marching strategy. We establish the relationship between the $L^2$-error bound and the $L^{\infty }/H^2$-stabilities of the numerical solution. Then, by converting the numerical schemes to a form compatible with the original format of the Allen–Cahn equation, using mathematical induction, the superconvergence property of nonlinear terms, and the spectrum argument, the optimal error estimates that only depends on the low-order polynomial degree of ${𝜖}^{-1}$ instead of $e^{T/{𝜖}^2}$ for both of the semi and fully-discrete schemes are derived. Numerical experiment also validates our theoretical convergence analysis.

A fully discrete semi-convex-splitting finite-element scheme with stabilization for a Cahn–Hilliard cross-diffusion system is analyzed. The system consists of parabolic fourth-order equations for the volume fraction of the fiber phase and solute concentration, modeling pre-patterning of lymphatic vessel morphology. The existence of discrete solutions is proved, and it is shown that the numerical scheme is energy stable up to stabilization, conserves the solute mass, and preserves the lower and upper bounds of the fiber phase fraction. Numerical experiments in two space dimensions using FreeFem illustrate the phase segregation and pattern formation.

In this paper, we consider integrating the scalar auxiliary variable time discretization with the virtual element method spatial discretization to obtain energy-stable schemes for Allen–Cahn-type gradient flow problems. In order to optimize CPU time during calculations, we propose two step-by-step solving SAV algorithms by introducing a novel auxiliary variable to replace the original one. Then, linear, decoupled, and unconditionally energy-stable numerical schemes are constructed. However, due to truncation errors, the auxiliary variable is not equivalent to the continuous case in the original definition. Therefore, we propose a novel relaxation technique to preserve the original energy dissipation rule. It not only retains all the advantages of the above algorithms but also improves accuracy and consistency. Finally, a series of numerical experiments are conducted to demonstrate the effectiveness of our method.

In this article, we overview recent developments of modern computational methods for the approximate solution of phase-field problems. The main difficulty for developing a numerical method for phase field equations is a severe stability restriction on the time step due to nonlinearity and high order differential terms. It is known that the phase field models satisfy a nonlinear stability relationship called gradient stability, usually expressed as a time-decreasing free-energy functional. This property has been used recently to derive numerical schemes that inherit the gradient stability. The first part of the article will discuss implicit-explicit time discretizations which satisfy the energy stability. The second part is to discuss time-adaptive strategies for solving the phase-field problems, which is motivated by the observation that the energy functionals decay with time smoothly except at a few critical time levels. The classical operator-splitting method is a useful tool in time discrtization. In the final part, we will provide some preliminary results using operator-splitting approach.