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High-order finite difference methods are efficient, easy to program, scales well in multiple dimensions and can be modified locally for various reasons (such as shock treatment for example). The main drawback have been the complicated and…
We use high order finite difference methods to solve the wave equation in the second order form. The spatial discretization is performed by finite difference operators satisfying a summation-by-parts property. The focus of this work is on…
We construct new, efficient, and accurate high-order finite differencing operators which satisfy summation by parts. Since these operators are not uniquely defined, we consider several optimization criteria: minimizing the bandwidth, the…
Preserving scalar boundedness is important for numerical schemes used in turbulent compressible multi-component flow simulations to prevent unphysical results and unstable simulations. However, ensuring scalar boundedness for high-order,…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
Finite difference schemes for the simulation of elastic waves in materi- als with jump discontinuities are presented. The key feature is the highly accurate treatment of interfaces where media discontinuities arise. The schemes are…
In the past decades, the finite difference methods for space fractional operators develop rapidly; to the best of our knowledge, all the existing finite difference schemes, including the first and high order ones, just work on uniform…
We develop summation by parts (SBP) approach for generating high-order finite-difference schemes on the interval and propose new sets of schemes up to the 12th order. The coefficients of the schemes are governed by values of grid spacing…
We derive high-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids. The schemes are fourth-order accurate in space and second-order accurate in time for vanishing correlation. In…
Multidimensional population balance models (PBMs) describe chemical and biological processes having a distribution over two or more intrinsic properties (such as size and age, or two independent spatial variables). The incorporation of…
In this paper, we design high order accurate and stable finite difference schemes for the initial-boundary value problem, associated with the magnetic induction equation with resistivity. We use Summation-By-Parts (SBP) finite difference…
The convergence rate of a multigrid method depends on the properties of the smoother and the so-called grid transfer operator. In this paper we define and analyze new grid transfer operators with a generic cutting size which are applicable…
We describe high order accurate and stable finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity…
We describe high order accurate and stable fully-discrete finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a…
It is well known that multigrid methods are optimally efficient for solution of elliptic equations (O(N)), which means that effort is proportional to the number of points at which the solution is evaluated). Thus this is an ideal method to…
In ordinary turbulence research it has been a long standing tradition to solve the equations in spectral space giving the best possible accuracy. This is indeed a natural choice for incompressible problems with periodic boundaries, but it…
In this paper, we consider finite difference approximations of the second order wave equation. We use finite difference operators satisfying the summation-by-parts property to discretize the equation in space. Boundary conditions and grid…
We develop a hybrid spatial discretization for the wave equation in second order form, based on high-order accurate finite difference methods and discontinuous Galerkin methods. The hybridization combines computational efficiency of finite…
Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the one-dimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for…
I prove that a centre manifold approach to creating finite difference models will consistently model linear dynamics as the grid spacing becomes small. Using such tools of dynamical systems theory gives new assurances about the quality of…