• Abstract

Coronary artery disease is a devastating complication of some patients undergoing liver transplantation. Anesthesia, anhepatic blood flow occlusion, and reperfusion of the liver can cause severe fluctuations in hemodynamics. However, the vast majority of liver transplant patients cannot undergo invasive coronary examinations due to their critical illness and abnormal coagulation function. In this paper, we present a retrospective case of acute myocardial infarction during surgery in order to demonstrate a noninvasive method to obtain coronary hemodynamic functional information based on scalable computational fluid dynamics technology. A $P_1-P_1$ stabilized finite element method and second-order backward differentiation formula are applied to discretize the time-dependent Navier-Stokes equations in the spatial and temporal directions, respectively. A Windkessel model constructed based on the measured clinic data is used to characterize the outlet blood flow. We then apply a parallel Newton-Krylov method with a restricted additive Schwarz preconditioner to accelerate the timeliness of the simulation. The simulated functional indicator successfully verifies the myocardial ischemia in the anhepatic phase of liver transplantation. We also present the parallel performance of the algorithm on a supercomputer, and the results show that the proposed solver achieves over $55\%$ parallel efficiency with $3840$ processor cores.

• Abstract

A kind of conservative upwind method is discussed for chemical oil recovery displacement in porous media. The mathematical model is formulated by a nonlinear convection-diffusion system dependent on the pressure, Darcy velocity, concentration and saturations. The flow equation is solved by a conservative block-centered method, and the pressure and Darcy velocity are obtained at the same time. The concentration and saturations are determined by convection-dominated diffusion equations, so an upwind approximation is adopted to eliminate numerical dispersion and nonphysical oscillation. Block-centered method is conservative locally. An upwind method with block-centered difference is used for computing the concentration. The saturations of different components are solved by the method of upwind fractional step difference, and the computational work is shortened significantly by dividing a three-dimensional problem into three successive one-dimensional problems and using the method of speedup. Using the variation discussion, energy estimates, the method of duality, and the theory of a priori estimates, we complete numerical analysis. Finally, numerical tests are given for showing the computational accuracy, efficiency and practicability of our approach.

• Abstract

A novel canonical Euler splitting method is proposed for nonlinear composite stiff functional differential-algebraic equations, the stability and convergence of the method is evidenced, theoretical results are further confirmed by some numerical experiments. Especially, the numerical method and its theories can be applied to special cases, such as delay differential-algebraic equations and integral differential-algebraic equations.

• Abstract

This paper presents the numerical solution of the time-dependent Gross-Pitaevskii Equation describing the movement of quantum mechanics particles under non-homogeneous boundary conditions. Due to their inherent non-linearity, the equation generally can not be solved analytically. Instead, a highly accurate approximation to the solutions defined in a finite domain is proposed, using the Crank-Nicolson difference method and Sinc Collocation numerical methods to discretize separately in time and space. Two Sinc numerical approaches, involving the Sinc Collocation Method (SCM) and the Sinc Derivative Collocation Method (SDCM), are easy to implement. The results demonstrate that Sinc numerical methods are highly efficient and yield accurate results. Mainly, the SDCM decays errors faster than the SCM. Future work suggests that the SDCM can be extensively applied to approximate solutions under other boundary conditions to demonstrate its broad applicability further.

• Abstract

In this paper, the piecewise spectral-collocation method is used to solve the second-order Volterra integral differential equation with nonvanishing delay. In this collocation method, the main discontinuity point of the solution of the equation is used to divide the partitions to overcome the disturbance of the numerical error convergence caused by the main discontinuity of the solution of the equation. Derivative approximation in the sense of integral is constructed in numerical format, and the convergence of the spectral collocation method in the sense of the $L^{\infty}$ and $L^{2}$ norm is proved by the Dirichlet formula. At the same time, the error convergence also meets the effect of spectral accuracy convergence. The numerical experimental results are given at the end also verify the correctness of the theoretically proven results.

• Abstract

In this paper, we construct a Crank-Nicolson finite volume element scheme and a two-grid decoupling algorithm for solving the time-dependent Schrodinger equation. Combining the idea of two-grid discretization, the decoupling algorithm involves solving a small coupling system on a coarse grid space and a decoupling system with two independent Poisson problems on a fine grid space, which can ensure the accuracy while the size of coarse grid is much coarser than that of fine grid. We further provide the optimal error estimate of these two schemes rigorously by using elliptic projection operator. Finally, numerical simulations are provided to verify the correctness of the theoretical analysis.

• Abstract

Numerical study on dynamic hydroelastic problems is usually rather complex due to the coupling of fluid and solid mechanics. Here, we demonstrate that the performance of a hydroelastic microfluidic oscillator can be analyzed using a simple equivalent circuit model. Previous studies reveal that its transition from the steady state to the oscillation state follows the negative-differential-resistance (NDR) mechanism. The performance is mainly determined by a bias fluidic resistor, and a pressure-variant resistor which further relates to the bending stiffness of the elastic diaphragm and the depth of the oscillation chamber. In this work, a numerical study is conducted to examine the effects of key design factors on the device robustness, the applicable fluid viscosity, the flow rate, and the transition pressure. The underlying physics is interpreted, providing a new perspective on hydroelastic oscillation problems. Relevant findings also provide design guidelines of the NDR fluidic oscillator.

• Abstract

In this paper, a three-dimensional time-dependent nonlinear Riesz space-fractional reaction-diffusion equation is considered. First, a linearized finite volume method, named BDF-FV,  is developed and analyzed via the discrete energy method, in which the space-fractional derivative is discretized by the finite volume element method and the time derivative is treated by the  backward differentiation formulae (BDF). The method is rigorously proved to be convergent with second-order accuracy both in time and space with respect to the discrete and continuous $L^2$ norms. Next, by adding high-order perturbation terms in time to the BDF-FV scheme, an alternating direction implicit linear finite volume scheme, denoted as BDF-FV-ADI, is proposed. Convergence with second-order accuracy is also strictly proved under a rough temporal-spatial stepsize constraint. Besides, efficient implementation of the ADI method is briefly discussed, based on a fast conjugate gradient (FCG) solver for the resulting symmetric positive definite linear algebraic systems.  Numerical experiments are presented to support the theoretical analysis and demonstrate the effectiveness and efficiency of the method for large-scale modeling and simulations.

• Abstract

In this paper, a two-grid block-centered finite difference method for the incompressible miscible displacement in porous medium is introduced and analyzed, which  is to solve a nonlinear equation on coarse mesh space of size $ H $ and a linear equation on fine grid of size $ h $. We establish the full discrete two-grid block-centered finite difference scheme on a uniform grid. The error estimates for the pressure, Darcy velocity, concentration variables are derived, which show that the discrete $L_2$ error is $\mathcal{O}(\Delta t+h^2+H^4)$. Finally, two numerical examples are provided to demonstrate the effectiveness and accuracy of our algorithm.

• Abstract

In this paper, the local domain-free discretization (DFD) method is extended to large eddy simulation (LES) of fluid-structure interaction and the vortex-induced vibration (VIV) of an elastically mounted rigid circular cylinder, which is held in the middle of a straight channel, is numerically investigated. The wall model based on the simplified turbulent boundary layer equations is employed to alleviate the requirement of mesh resolution in the near-wall region. The ability of the method for fluid-structure interaction is demonstrated by simulating flows over a circular cylinder undergoing VIV. The cylinder is neutrally buoyant with a reduced mass ${{m}^{\ast}}=11$ and has a low damping ratio $\zeta =0.001$. The numerical experiment of the VIV of a cylinder in an unbounded flow shows that the present LES-DFD method is more accurate and reliable than the referenced RANS and DES methods. For the cylinder in the middle of a straight channel, the effect of the channel height (${{d}^{\ast}}=d/D$) is investigated. The variations of the response amplitude, vortex-shedding pattern and the length of the induced separation zone in the channel boundary layers with the channel height are presented.

• Abstract

In this paper, we propose and analyze a second order accurate in time, mass lumped mixed finite element scheme for the Cahn-Hilliard equation with a logarithmic Flory-Huggins energy potential. The standard backward differentiation formula (BDF) stencil is applied in the temporal discretization. In the chemical potential approximation, both the logarithmic singular terms and the surface diffusion term are treated implicitly, while the expansive term is explicitly updated via a second-order Adams-Bashforth extrapolation formula, following the idea of the convex-concave decomposition of the energy functional. In addition, an artificial Douglas-Dupont regularization term is added to ensure the energy dissipativity. In the spatial discretization, the mass lumped finite element method is adopted. We provide a theoretical justification of the unique solvability of the mass lumped finite element scheme, using a piecewise linear element. In particular, the positivity is always preserved for the logarithmic arguments in the sense that the phase variable is always located between $-1$ and $1$. In fact, the singular nature of the implicit terms and the mass lumped approach play an essential role in the positivity preservation in the discrete setting. Subsequently, an unconditional energy stability is proven for the  proposed numerical scheme. In addition, the  convergence analysis and error estimate of the numerical scheme are also presented. Two numerical experiments are carried out to verify the theoretical properties.