A novel Douglas alternating direction implicit (ADI) method is proposed in this work to solve a two-dimensional (2D) heat equation with interfaces. The ADI scheme is a powerful finite difference method for solving parabolic equations, due to its unconditional stability and high efficiency. However, it suffers from a serious accuracy reduction in space for interface problems with different mater...

Summation-by-parts (SBP) finite-difference discretizations share many attractive properties with Galerkin finite-element methods (FEMs), including time stability and superconvergent functionals; however, unlike FEMs, SBP operators are not completely determined by a basis, so the potential exists to tailor SBP operators to meet different objectives. To date, application of highorder SBP discreti...

We present a fourth-order compact finite difference scheme on the face centered cubic (FCC) grids for the numerical solution of the two-dimensional convection diffusion equation. The seven-point formula is defined on a regular hexagon, where the strategy of directional derivative is employed to make the derivation procedure straightforward, efficient, and concise. A corresponding multigrid meth...

An unconditionally stable finite difference scheme for systems whose dynamics are described by a fourth-order partial differential equation is developed with the use of a regular hexagonal grid. The scheme is motivated by the well-known Crank-Nicolson discretization that was originally developed for second-order systems and it is used to develop a discrete in time and space model of a deformabl...

We use the local Fourier analysis to determine the properties of the multigrid method when used in modeling the skin penetration of a drug. The analyses of these properties can be very in designing an efficient structure of the multigrid method and in comparing the element and finite difference discretization techniques. After the theoretical results obtained, we also present some numerical res...

Parameter identification problems for partial differential equations usually lead to nonlinear inverse problems. A typical property of such problems is their instability, which requires regularization techniques, like, e.g., Tikhonov regularization. The main focus of this paper will be on efficient methods for determining a suitable regularization parameter by using adaptive finite element disc...

Abstract A nonlinear conjugate gradient method is derived for the inverse problem of identifying a treatment parameter in model reaction–diffusion type corresponding to evolution brain tumours under therapy. The reconstructed from additional information about tumour taken at fixed instance time. Well-posedness direct problems used iterative outlined as well uniqueness solution problem. Moreover...

The Wigner formalism provides a convenient formulation of quantum mechanics in the phase space. Deterministic solutions of the Wigner equation are especially needed for problems where phase space quantities vary over several orders of magnitude and thus can not be resolved by the existing stochastic approaches. However, finite difference schemes have been problematic due to the discretization o...

We develop discontinuous Galerkin framework for solving direct and inverse problems in fluorescence diffusion optical tomography in turbid media. We show the advantages and the disadvantages of this method by comparing it with previously developed framework based on the finite volume discretization. The reconstruction algorithm was used with time-gated experimental dataset acquired by imaging a...

The main concern of this paper is with the stable discretisation of linear partial differential equations of evolution with time-varying coefficients. We commence by demonstrating that an approximation of the first derivative by a skew-symmetric matrix is fundamental in ensuring stability for many differential equations of evolution. This motivates our detailed study of skew-symmetric different...