We propose a stable sixth-order compact finite difference scheme coupled with a fifth-order staggered boundary scheme and the Runge-Kutta adaptive time stepping based on 3(2) Bogacki-Shampine pairs for pricing American options. To compute the free-boundary simultaneously and precisely with the option value and Greeks, we introduce a logarithmic Landau transformation and then remove the convective term in the pricing model by introducing the delta sensitivity, so that an efficient sixth-order compact scheme can be easily implemented. The main challenge in coupling the sixth order compact scheme in discrete form is to efficiently account for the near-boundary scheme. In this study, we introduce novel fifth and sixth-order Dirichlet near-boundary schemes that are suitable for solving our model. The optimal exercise boundary and other boundary values are approximated using a high-order analytical approximation that is obtained from a novel fifth-order staggered boundary scheme. Furthermore, we investigate the smoothness of the first and second-order derivatives of the optimal exercise boundary which is obtained from the high-order analytical approximation. Coupled with the adaptive time integration method, the interior values are then approximated using the sixth order compact schemes. As such, the expected convergence rate is reasonably achieved, and the present numerical scheme is very fast in computation and gives highly accurate solutions with very coarse grids.

We propose a high order unfitted finite element method for solving time-harmonic Maxwell interface problems. The unfitted finite element method is based on a mixed formulation in the discontinuous Galerkin framework on a Cartesian mesh with possible hanging nodes. The $H^2$ regularity of the solution to Maxwell interface problems with $C^2$ interfaces in each subdomain is proved. Practical interface-resolving mesh conditions are introduced under which the $hp$ inverse estimates on three-dimensional curved domains are proved. Stability and $hp$ a priori error estimate of the unfitted finite element method are proved. Numerical results are included to illustrate the performance of the method.

This article proposes and analyzes a rotational pressure-correction method for the Navier-Stokes/Darcy (NSD) system with Beavers-Joseph-Saffman-Jones interface conditions. This method mainly solves the Navier-Stokes/Darcy problem in two steps. The first step is the viscous step. The intermediate velocity can be obtained after the pressure gradient is explicitly processed by the algorithm. The second step is the projection step, which first projects the intermediate velocity onto a divergence-free space, and then corrects the velocity and pressure. The main advantage of these methods is that they have first/second order accuracy and do not have the incompressibility constraint of NSD system. For solving the Navier-Stokes equations, each time step requires only one vector-valued elliptic equation and one scalar-valued Poisson equation. Therefore, this method has high computational efficiency. Compared with other traditional related methods, this method is no longer affected by any artificial boundary conditions, and can achieve the optimal convergence order. Finally, unconditional stability and long time stability are established. 2D/3D numerical experiments are presented to illustrate the features of the proposed method and verify the results of the theoretical analysis.

Turbulent flows strain resources, both memory and CPU speed. A family of second-order, $G$-stable time-stepping methods proposed by Dahlquist, Liniger, and Nevanlinna (the DLN method) has great accuracy and allows large time steps, requiring less memory and fewer FLOPS. The DLN method can also be implemented adaptively. The classical Smagorinsky model, as an effective way to approximate a resolved mean velocity, has recently been corrected to represent a flow of energy from unresolved fluctuations to the resolved mean velocity. In this paper, we apply the DLN method to one corrected Smagorinsky model and provide a detailed numerical analysis of the stability and consistency. We prove that the numerical solutions under arbitrary time step sequences are unconditionally stable in the long term and converge in second order. We also provide error estimates under certain time-step conditions. Numerical tests are given to confirm the rate of convergence and also to show that the adaptive DLN algorithm helps to control numerical dissipation so that a flow of energy from unresolved fluctuations to the resolved mean velocity is visible.

We present a finite volume method preserving the invariant region property (IRP) for the reaction-diffusion systems with quasimonotone functions, including nondecreasing, decreasing, and mixed quasimonotone systems. The diffusion terms and time derivatives are discretized using a finite volume method that satisfies the discrete maximum principle (DMP) and the backward Euler method, respectively. The discretization leads to an implicit and nonlinear scheme, and it is proved to preserve the invariant region property unconditionally. We construct an iterative algorithm and prove the invariant region property at each iteration step. Numerical examples are provided to confirm the accuracy and invariant region property of our scheme.