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In this work, we use two numerical methods in order to solve the Black-Scholes equation with specified initial and boundary conditions.We use a classical finite difference scheme and an unconditionally positive definite scheme. The stability region is obtained in terms of an inequality involving the temporal step size. We compare the results from both methods at some different values of temporal step size by computing L1, L2 and L∞ errors. [ABSTRACT FROM AUTHOR]

In this paper, we devise three finite difference methods in order to solve a 1-D Korteweg-de-Vries Burger's (KdVB) equation with specified initial and boundary conditions. The test case is taken from [6] and the exact solution is known for this problem. We obtain the stability region and compare the performance of our three methods with the sinc-collocation method [6] by computing L∞, L2 errors at some different times and also obtain absolute errors and relative errors at some spatial nodes. [ABSTRACT FROM AUTHOR]

In this work, three finite difference methods have been used to solve a three dimensional advection-diffusion equation with given initial and boundary conditions. The three methods are fourth order finite difference method, Crank-Nicolson and Implicit Chapeau Function. We compare the performance of the methods by computing L2 error, L∞ error and some performance indices such as mass distribution ratio (MDR), mass conservation ratio (MCR), total mass and R² which is a measure of total variation in particle distribution. [ABSTRACT FROM AUTHOR]

An unconditionally positive definite scheme has been derived in [1] to approximate a linear advection-diffusion-reaction equation which models exponential travelling waves and the coefficients of advective, diffusive and reactive terms have been chosen as one. The scheme has been baptised as Unconditionally Positive Finite Difference (UPFD). In this work, we use the UPFD scheme to solve the advection-diffusion-reaction problem in [1] and we also extend our study to three other important regimes involved in this model. The temporal step size is varied while fixing the spatial step size. We compute some errors namely; L1 error, dispersion, dissipation errors. We also study the variation of the modulus of the exact amplification factor, modulus of amplification factor of the scheme and relative phase error, all vs the phase angle for the four different regimes. [ABSTRACT FROM AUTHOR]

In this paper, three numerical methods have been used to solve a 1-D Convection-Diffusion equation with specified initial and boundary conditions. The methods used are the third order upwind scheme [1], fourth order upwind scheme [1] and a Non-Standard Finite Difference (NSFD) scheme [4]. The problem we considered has steep boundary layers near x = 1 [3] and this is a challenging test case as many schemes are plagued by nonphysical oscillation near steep boundaries. We compute the L2 and L∞ errors, dissipation and dispersion errors when the three numerical schemes are used and observe that the NSFD is much better than the other two schemes for both coarse and fine grids and also at low and high Reynolds numbers. [ABSTRACT FROM AUTHOR]