We present high-order compact schemes for a linear second-order parabolic partial differential equation (PDE) with mixed second-order derivative terms in two spatial dimensions. The schemes are applied to option pricing PDE for a family of stochastic volatility models. We use a nonuniform grid with more grid-points around the strike price. The schemes are fourth-order accurate in space and seco...

: : In this paper, we present a new approach to resolve linear weakly-singular partial integro-differential equations by first removing the singularity using Taylor's approximation and transform the given partial integrodifferential equations into an partial differential equation. After that the fourth order compact finite difference scheme and collocation method is presented to obtain system o...

High-order simulation of flows containing shock waves is an extremely difficult task due to the discontinuous changes in flow properties across the shock. The present work investigates a shock-detecting sensor for filtering of high-order compact finitedifference schemes to examine the shock-capturing in direct simulation of NavierStokes solver. Based on the accuracy and minimum dissipation erro...

In this paper simple polynomial interpolation is used to derive arbitrarily high-order compact schemes for the first derivative and tridiagonal compact schemes for the second derivative (consisting of three second derivative nodes in the interior and two on the boundary) on non-uniform grids. Boundary and near boundary schemes of the same order as the interior are also developed using polynomia...

In fusion plasmas diffusion tensors are extremely anisotropic due to the high temperature and large magnetic field strength. This causes diffusion, heat conduction, and viscous momentum loss, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular directio...

where A : D(A) ⊆H →H , α : D(α) ⊆H →H , and β : D(β) ⊆H →H are maximal monotone operators in the real Hilbert space H (satisfying some specific properties), a, b are given elements in the domain D(A) of A, f ∈ L2(0,T ;H), and p,r : [0,T] → R are continuous functions, p(t) ≥ k > 0 for all t ∈ [0,T]. Particular cases of this problem were considered before in [9, 10, 12, 15, 16]. If p ≡ 1, r ≡ 0, ...

In this paper a new family of high-order finite-difference shock-capturing central schemes for hyperbolic systems with stiff source is presented. The schemes are based on recently developed finite difference discretization on staggered grids, coupled with implicit-explicit (IMEX) time discretization for an efficient treatment of the source term. Numerical tests show the robustness and accuracy ...

The finite-difference-time-domain (FDTD), although recognized as a flexible, robust and simple to implement method for solving complex electromagnetic problems, is subject to numerical dispersion errors. In addition to the traditional ways for reducing dispersion, i.e., increasing sampling rate and using higher order degrees of accuracy, a number of schemes have been proposed recently. In this ...