A Fitted L-Multi-Point Flux Approximation Method for Pricing Options

In this paper, we introduce a special kind of finite volume method called Multi-Point Flux Approximation method (MPFA) to price European and American options in two dimensional domain. We focus on the L-MPFA method for space discretization of the diffusion term of Black–Scholes operator. The degeneracy of the Black-Scholes operator is tackled using the fitted finite volume method. This combination of fitted finite volume method and L-MPFA method coupled to upwind methods gives us a novel scheme, called the fitted L-MPFA method. Numerical experiments show the accuracy of the novel fitted L-MPFA method comparing to well known schemes for pricing options.

Modeling and Simulation of Supersonic Flow in 2D Nozzle Using MacCormack’s and Upwind Methods

In this work, two-dimensional inviscid supersonic flow in nozzle has been investigated using CFD schemes and characteristics method. The employed scheme is MacCormack’s finite volume method. Our own code CHARMAC, was written using MATLAB environment. Standard boundary conditions and the grid parameters were considered to solve the problem. Before analyzing the flow by CFD method, we obtained the ideal nozzle geometry using the method of characteristics for a 2D divergent Nozzle. Using 2D nozzle flow relations, an optimal throat area is found. At the end we compare the results with the advection upstream splitting Method (Ausm) and Fluent.

Fast Simulation of Polymer Injection in Heavy-Oil Reservoirs on the Basis of Topological Sorting and Sequential Splitting

Summary We present a set of algorithms for sequential solution of flow and transport that can be used for efficient simulation of polymer injection modeled as a compressible two-phase system. Our formulation gives a set of nonlinear transport equations that can be discretized with standard implicit upwind methods to conserve mass and volume independent of the timestep. In the absence of gravity and capillary forces, the resulting nonlinear system of discrete transport equations can be permuted to lower triangular form with a simple topological-sorting method from graph theory. This gives an optimal nonlinear solver that computes the solution cell by cell with local iteration control. The single-cell systems can be reduced to a nested set of nonlinear scalar equations that can be bracketed and solved with standard gradient or root-bracketing methods. The resulting method gives orders-of-magnitude reduction in run times and increases the feasible timestep sizes. In fact, one can prove that the solver is unconditionally stable and will produce a solution for arbitrarily large timesteps. For cases with gravity, the same method can be applied as part of a nonlinear Gauss-Seidel method. Altogether, our results demonstrate that sequential splitting combined with single-point upwind discretizations can become a viable alternative to streamline methods for speeding up simulation of advection-dominated systems.