This paper is devoted to the American option pricing problem governed by the Black-Scholes equation. The existence of an optimal exercise policy makes the problem a free boundary value problem of a parabolic equation on an unbounded domain. The optimal exercise boundary satisfies a nonlinear Volterra integral equation and is solved by a high-order collocation method based on graded meshes. This free boundary is then deformed to a fixed boundary by the front-fixing transformation. The boundary condition at infinity (due to the fact that the underlying asset's price could be arbitrarily large in theory), is treated by the perfectly matched layer technique. Finally, the resulting initial-boundary value problems for the option price and some of the Greeks on a bounded rectangular space-time domain are solved by a finite element method. In particular, for Delta, one of the Greeks, we propose a discontinuous Galerkin method to treat the discontinuity in its initial condition. Convergence results for these two methods are analyzed and several numerical simulations are provided to verify these theoretical results.

In this paper we are concerned with numerical methods for nonlinear time-dependent problem coupled by electron, ion and photon temperatures in two dimensions, which is called the 2D-3T heat conduction equations. We propose discontinuous Galerkin (DG) methods for the discretization of the equations. For solving the resulting discrete system, we employ two domain decomposition (DD) preconditioners, one of which is associated with the non-overlapping DDM and the other is based on DDM with small overlap. The preconditioners are constructed by dropping the couplings between particles and each preconditioner consists of three preconditioners with smaller matrix size. To gauge the efficiency of the preconditioners, we test two examples and make different settings of parameters. Numerical results show that the proposed preconditioners are very effective to the 2D-3T problem.

In this work, we investigate the numerical approximation of the compressible Navier-Stokes equations under the framework of discontinuous Galerkin methods. For discretization of the viscous and heat fluxes, we extend and apply the symmetric direct discontinuous Galerkin (SDDG) method which is originally introduced for scalar diffusion problems. The original compressible Navier-Stokes equations are rewritten into an equivalent form via homogeneity tensors. Then, the numerical diffusive fluxes are constructed from the weak formulation of primal equations directly without converting the second-order equations to a first-order system. Additional numerical flux functions involving the jump of second order derivative of test functions are added to the original direct discontinuous Galerkin (DDG) discretization. A number of numerical tests are carried out to assess the practical performance of the SDDG method for the two dimensional compressible Navier-Stokes equations. These numerical results obtained demonstrate that the SDDG method can achieve the optimal order of accuracy. Especially, compared with the well-established symmetric interior penalty (SIP) method [18], the SDDG method can maintain the expected optimal order of convergence with a smaller penalty coefficient.

In magnetically confined plasma research, the understandings of small and large perturbations at equilibrium are both critical for plasma controlling and steady state operation. Numerical simulations using original MHD model can hardly give clear picture for small perturbations, while non-conservative perturbed MHD model may break conservation law, and give unphysical results when perturbations grow large after long-time computation. In this paper, we present a nonlinear conservative perturbed MHD model by splitting primary variables in original MHD equations into equilibrium part and perturbed part, and apply an approach in the framework of discontinuous Galerkin (DG) spatial discretization for numerical solutions. This enables high resolution of very small perturbations, and also gives satisfactory non-smooth solutions for large perturbations, which are both broadly concerned in magnetically confined plasma research. Numerical examples demonstrate satisfactory performance of the proposed model clearly. For small perturbations, the results have higher resolution comparing with the original MHD model; for large perturbations, the non-smooth solutions match well with existing references, confirming reliability of the model for instability investigations in magnetically confined plasma numerical research.

We present a high-order discontinuous Galerkin (DG) method for the time domain Maxwell's equations in three-dimensional heterogeneous media. New hierarchical orthonormal basis functions on unstructured tetrahedral meshes are used for spatial discretization while Runge-Kutta methods for time discretization. A uniaxial perfectly matched layer (UPML) is employed to terminate the computational domain. Exponential convergence with respect to the order of the basis functions is observed and large parallel speedup is obtained for a plane-wave scattering model. The rapid decay of the out-going wave in the UPML is shown in a dipole radiation simulation. Moreover, the low frequency electromagnetic fields excited by a horizontal electric dipole (HED) and a vertical magnetic dipole (VMD) over a layered conductive half-space and a high frequency ground penetrating radar (GPR) detection for an underground structure are investigated, showing the high accuracy and broadband simulation capability of the proposed method.

Offline computation is an essential component in most multiscale model reduction techniques. However, there are multiscale problems in which offline procedure is insufficient to give accurate representations of solutions, due to the fact that offline computations are typically performed locally and global information is missing in these offline information. To tackle this difficulty, we develop an online local adaptivity technique for local multiscale model reduction problems. We design new online basis functions within Discontinuous Galerkin method based on local residuals and some optimally estimates. The resulting basis functions are able to capture the solution efficiently and accurately, and are added to the approximation iteratively. Moreover, we show that the iterative procedure is convergent with a rate independent of physical scales if the initial space is chosen carefully. Our analysis also gives a guideline on how to choose the initial space. We present some numerical examples to show the performance of the proposed method.

In this paper we consider a discontinuous Galerkin discretization of the ideal magnetohydrodynamics (MHD) equations on unstructured meshes, and the divergence free constraint (∇·B=0) of its magnetic field B. We first present two approaches for maintaining the divergence free constraint, namely the approach of a locally divergence free projection inspired by locally divergence free elements [19], and another approach of the divergence cleaning technique given by Dedner et al. [15]. By combining these two approaches we obtain a efficient method at the almost same numerical cost. Finally, numerical experiments are performed to show the capacity and efficiency of the scheme.

In this paper, a new combined method is presented to simulate saltwater intrusion problem. A splitting positive definite mixed element method is used to solve the water head equation, and a symmetric discontinuous Galerkin (DG) finite element method is used to solve the concentration equation. The introduction of these two numerical methods not only makes the coefficient matrixes symmetric positive definite, but also does well with the discontinuous problem. The convergence of this method is considered and the optimal L2-norm error estimate is also derived.

This paper concerns numerical computation of a fourth order eigenvalue problem. We first show the well-posedness of the source problem. An interior penalty discontinuous Galerkin method (C0IPG) using Lagrange elements is proposed and its convergence is studied. The method is then used to compute the eigenvalues. We show that the method is spectrally correct and prove the optimal convergence. Numerical examples are presented to validate the theory.

In order to suppress the failure of preserving positivity of density or pressure, a positivity-preserving limiter technique coupled with h-adaptive Runge-Kutta discontinuous Galerkin (RKDG) method is developed in this paper. Such a method is implemented to simulate flows with the large Mach number, strong shock/obstacle interactions and shock diffractions. The Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is also presented. This approach directly uses the cell solution polynomial of DG finite element space as the interpolation formula. The method is validated by the well documented test examples involving unsteady compressible flows through complex bodies over a large Mach numbers. The numerical results demonstrate the robustness and the versatility of the proposed approach.

In this paper, a DG (Discontinuous Galerkin) method which has been widely employed in CFD (Computational Fluid Dynamics) is used to solve the two-dimensional time-domain Maxwell’s equations for complex geometries on unstructured mesh. The element interfaces on solid boundary are treated in both curved way and straight way. Numerical tests are performed for both benchmark problems and complex cases with varying orders on a series of grids, where the high-order convergence in accuracy can be observed. Both the curved and the straight solid boundary implementation can give accurate RCS (Radar Cross-Section) results with sufficiently small mesh size, but the curved solid boundary implementation can significantly improve the accuracy when using relatively large mesh size. More importantly, this CFD-based high-order DG method for the Maxwell’s equations is very suitable for complex geometries.

The staggered discontinuous Galerkin (SDG) method has been recently developed for the numerical approximation of partial differential equations. An important advantage of such methodology is that the numerical solution automatically satisfies some conservation properties which are also satisfied by the exact solution. In this paper, we will consider the numerical approximation of the inviscid Burgers equation by the SDG method. For smooth solutions, we prove that our SDG method has the properties of mass and energy conservation. It is well-known that extra care has to be taken at locations of shocks and discontinuities. In this respect, we propose a local total variation (TV) regularization technique to suppress the oscillations in the numerical solution. This TV regularization is only performed locally where oscillation is detected, and is thus very efficient. Therefore, the resulting scheme will preserve the mass and energy away from the shocks and the numerical solution is regularized locally near shocks. Detailed description of the method and numerical results are presented.

Discontinuous Galerkin (DG) and matrix-free finite element methods with a novel projective pressure estimation are combined to enable the numerical modeling of magma dynamics in 2D and 3D using the library deal.II. The physical model is an advection-reaction type system consisting of two hyperbolic equations to evolve porosity and soluble mineral abundance at local chemical equilibrium and one elliptic equation to recover global pressure. A combination of a discontinuous Galerkin method for the advection equations and a finite element method for the elliptic equation provide a robust and efficient solution to the channel regime problems of the physical system in 3D. A projective and adaptively applied pressure estimation is employed to significantly reduce the computational wall time without impacting the overall physical reliability in the modeling of important features of melt segregation, such as melt channel bifurcation in 2D and 3D time dependent simulations.

In this paper, we present a superconvergence result for the bi-k degree time-space fully discontinuous finite element of first-order hyperbolic problems. Based on the element orthogonality analysis (EOA), we first obtain the optimal convergence order of discontinuous Galerkin finite element solution. Then we use orthogonality correction technique to prove a superconvergence result at right Radau points, which is higher one order than the optimal convergence rate. Finally, numerical results are presented to illustrate the theoretical analysis.

The pricing of option contracts is one of the classical problems in Mathematical Finance. While useful exact solution formulas exist for simple contracts, typically numerical simulations are mandated due to the fact that standard features, such as early-exercise, preclude the existence of such solutions. In this paper we consider derivatives which generalize the classical Black-Scholes setting by not only admitting the early-exercise feature, but also considering assets which evolve by the Constant Elasticity of Variance (CEV) process (which includes the Geometric Brownian Motion of Black-Scholes as a special case). In this paper we investigate a Discontinuous Galerkin method for valuing European and American options on assets evolving under the CEV process which has a number of advantages over existing approaches including adaptability, accuracy, and ease of parallelization.

A Newton/LU-SGS (lower-upper symmetric Gauss-Seidel) iteration implicit method was developed to solve two-dimensional Euler and Navier-Stokes equations by the DG/FV hybrid schemes on arbitrary grids. The Newton iteration was employed to solve the nonlinear system, while the linear system was solved with LU-SGS iteration. The effect of several parameters in the implicit scheme, such as the CFL number, the Newton sub-iteration steps, and the update frequency of Jacobian matrix, was investigated to evaluate the performance of convergence history. Several typical test cases were simulated, and compared with the traditional explicit Runge-Kutta (RK) scheme. Firstly the Couette flow was tested to validate the order of accuracy of the present DG/FV hybrid schemes. Then a subsonic inviscid flow over a bump in a channel was simulated and the effect of parameters was investigated also. Finally, the implicit algorithm was applied to simulate a subsonic inviscid flow over a circular cylinder and the viscous flow in a square cavity. The numerical results demonstrated that the present implicit scheme can accelerate the convergence history efficiently. Choosing proper parameters would improve the efficiency of the implicit scheme. Moreover, in the same framework, the DG/FV hybrid schemes are more efficient than the same order DG schemes.

In [SIAM J. Sci. Comput., 35(2)(2013), A1049-A1072], a class of multi-domain hybrid DG and WENO methods for conservation laws was introduced. Recent applications of this method showed that numerical instability may encounter if the DG flux with Lagrangian interpolation is applied as the interface flux during the moment of conservative coupling. In this continuation paper, we present a more robust approach in the construction of DG flux at the coupling interface by using WENO procedures of reconstruction. Based on this approach, such numerical instability is overcome very well. In addition, the procedure of coupling a DG method with a WENO-FD scheme on hybrid meshes is disclosed in detail. Typical testing cases are employed to demonstrate the accuracy of this approach and the stability under the flexibility of using either WENO-FD flux or DG flux at the moment of requiring conservative coupling.

We study the enhancement of accuracy, by means of the convolution post-processing technique, for discontinuous Galerkin(DG) approximations to hyperbolic problems. Previous investigations have focused on the superconvergence obtained by this technique for elliptic, time-dependent hyperbolic and convection-diffusion problems. In this paper, we demonstrate that it is possible to extend this post-processing technique to the hyperbolic problems written as the Friedrichs’ systems by using an upwind-like DG method. We prove that the L2-error of the DG solution is of order k+1/2, and further the post-processed DG solution is of order 2k+1 if Qk-polynomials are used. The key element of our analysis is to derive the (2k+1)-order negative norm error estimate. Numerical experiments are provided to illustrate the theoretical analysis.

A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell’s equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in our scheme, discontinuous Galerkin methods are used to discretize not only the spatial domain but also the temporal domain. The proposed numerical scheme is proved to be unconditionally stable, and a convergent rate is established under the L2-norm when polynomials of degree atmost r and k are used for temporal and spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order in time step is observed numerically for the numerical fluxes w.r.t. temporal variable at the grid points.

In this paper, we propose and analyze the interior penalty discontinuous Galerkin method for H(div)-elliptic problem. An optimal a priori error estimate in the energy norm is proved. In addition, a residual-based a posteriori error estimator is obtained. The estimator is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to demonstrate the effectiveness of our method.