Search results for "finite differences"
showing 6 items of 6 documents
Long-range cohesive interactions of non-local continuum faced by fractional calculus
2008
Abstract A non-local continuum model including long-range forces between non-adjacent volume elements has been studied in this paper. The proposed continuum model has been obtained as limit case of two fully equivalent mechanical models: (i) A volume element model including contact forces between adjacent volumes as well as long-range interactions, distance decaying, between non-adjacent elements. (ii) A discrete point-spring model with local springs between adjacent points and non-local springs with distance-decaying stiffness connecting non-adjacent points. Under the assumption of fractional distance-decaying interactions between non-adjacent elements a fractional differential equation in…
Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation
2004
A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.
Scheduled Relaxation Jacobi method: improvements and applications
2016
Elliptic partial differential equations (ePDEs) appear in a wide variety of areas of mathematics, physics and engineering. Typically, ePDEs must be solved numerically, which sets an ever growing demand for efficient and highly parallel algorithms to tackle their computational solution. The Scheduled Relaxation Jacobi (SRJ) is a promising class of methods, atypical for combining simplicity and efficiency, that has been recently introduced for solving linear Poisson-like ePDEs. The SRJ methodology relies on computing the appropriate parameters of a multilevel approach with the goal of minimizing the number of iterations needed to cut down the residuals below specified tolerances. The efficien…
NUMERICAL SIMULATION OF MAGNETIC DROPLET DYNAMICS IN A ROTATING FIELD
2013
Dynamics and hysteresis of an elongated droplet under the action of a rotating magnetic field is considered for mathematical modelling. The shape of droplet is found by regularization of the ill-posed initial–boundary value problem for nonlinear partial differential equation (PDE). It is shown that two methods of the regularization – introduction of small viscous bending torques and construction of monotonous continuous functions are equivalent. Their connection with the regularization of the ill-posed reverse problems for the parabolic equation of heat conduction is remarked. Spatial discretization is carried out by the finite difference scheme (FDS). Time evolution of numerical solutions …
Mathematical modelling of an elongated magnetic droplet in a rotating magnetic field
2012
Dynamics of an elongated droplet under the action of a rotating magnetic field is considered by mathematical modelling. The actual shape of a droplet is obtained by solving the initial-boundary value problem of a nonlinear singularly perturbed partial differential equation (PDE). For the discretization in space the finite difference scheme (FDS) is applied. Time evolution of numerical solutions is obtained with MATLAB by solving a large system of ordinary differential equations (ODE).
An order-adaptive compact approximation Taylor method for systems of conservation laws
2021
Abstract We present a new family of high-order shock-capturing finite difference numerical methods for systems of conservation laws. These methods, called Adaptive Compact Approximation Taylor (ACAT) schemes, use centered ( 2 p + 1 ) -point stencils, where p may take values in { 1 , 2 , … , P } according to a new family of smoothness indicators in the stencils. The methods are based on a combination of a robust first order scheme and the Compact Approximate Taylor (CAT) methods of order 2p-order, p = 1 , 2 , … , P so that they are first order accurate near discontinuities and have order 2p in smooth regions, where ( 2 p + 1 ) is the size of the biggest stencil in which large gradients are n…