A. R. Appadu

Numerical Solution of the 1D Advection-Diffusion Equation Using Standard and Nonstandard Finite Difference Schemes

Three numerical methods have been used to solve the one-dimensional advection-diffusion equation with constant coefficients. This partial differential equation is dissipative but not dispersive. We consider the Lax-Wendroff scheme which is explicit, the Crank-Nicolson scheme which is implicit, and a nonstandard finite difference scheme (Mickens 1991). We solve a 1D numerical experiment with specified initial and boundary conditions, for which the exact solution is known using all these three schemes using some different values for the space and time step sizes denoted by and , respectively, for which the Reynolds number is 2 or 4. Some errors are computed, namely, the error rate with respect to the norm, dispersion, and dissipation errors. We have both dissipative and dispersive errors, and this indicates that the methods generate artificial dispersion, though the partial differential considered is not dispersive. It is seen that the Lax-Wendroff and NSFD are quite good methods to approximate the 1D advection-diffusion equation at some values of and . Two optimisation techniques are then implemented to find the optimal values of when for the Lax-Wendroff and NSFD schemes, and this is validated by numerical experiments.

Investigating the shock-capturing properties of some composite numerical schemes for the 1-D linear advection equation

This paper enables us to understand better the shock-capturing property of composite schemes. The study allows us to understand why not all composite schemes can be effective to control dispersion and dissipation in regions of shocks when used to solve 1-D linear advection problems. We use a technique called the Minimised Integrated Exponential Error for Low Dispersion and Low Dissipation to analyse the shock-capturing property of some selected numerical methods applied to the 1-D linear advection equation. This technique also allows us to obtain the optimal cfl number. Numerical experiments are performed and the errors are quantified into dispersion and dissipation using a technique devised by Takacs. It is seen that the errors are dependent on the cfl number used. At the optimal cfl, we observe that the dispersion and dissipation errors are least as compared to other cfl numbers.

Comparison of some optimisation techniques for numerical schemes discretising equations with advection terms

We have considered the measures of errors devised by Tam and Webb (1993), Bogey and Bailly (2002) and by Berland et al. (2007) to construct low dispersion, low dissipation and high order numerical schemes in computational aeroacoustics. We modify their measures to obtain three different techniques of optimisation. We investigate the strength and weak points of these three optimisation techniques together with our technique of minimised integrated exponential error for low dispersion and low dissipation (MIEELDLD) (Appadu and Dauhoo, 2009, 2011; Appadu, 2011) when controlling the grade and balance of dispersion and dissipation with reference to some numerical schemes applied to the 1D linear advection equation and the Korteweg-de-Vries Burgers equations. It is observed that the technique of MIEELDLD is more effective than the other three techniques to control the grade and balance of dissipation and dispersion in numerical schemes. Therefore, we use MIEELDLD to optimise parameters in the α-scheme and to obtain a modification to the beam-warming scheme which has improved shock capturing properties. Some numerical experiments are carried out to validate the results by showing that when the optimal parameters are used, better results in terms of shock-capturing properties are obtained.

Optimized Weighted Essentially Nonoscillatory Third-Order Schemes for Hyperbolic Conservation Laws

We describe briefly how a third-order Weighted Essentially Nonoscillatory (WENO) scheme is derived by coupling a WENO spatial discretization scheme with a temporal integration scheme. The scheme is termed WENO3. We perform a spectral analysis of its dispersive and dissipative properties when used to approximate the 1D linear advection equation and use a technique of optimisation to find the optimal cfl number of the scheme. We carry out some numerical experiments dealing with wave propagation based on the 1D linear advection and 1D Burger’s equation at some different cfl numbers and show that the optimal cfl does indeed cause less dispersion, less dissipation, and lower errors. Lastly, we test numerically the order of convergence of the WENO3 scheme.

The Technique of MIEELDLD in Computational Aeroacoustics

The numerical simulation of aeroacoustic phenomena requires high-order accurate numerical schemes with low dispersion and low dissipation errors. A technique has recently been devised in a Computational Fluid Dynamics framework which enables optimal parameters to be chosen so as to better control the grade and balance of dispersion and dissipation in numerical schemes (Appadu and Dauhoo, 2011; Appadu, 2012a; Appadu, 2012b; Appadu, 2012c). This technique has been baptised as the Minimized Integrated Exponential Error for Low Dispersion and Low Dissipation (MIEELDLD) and has successfully been applied to numerical schemes discretising the 1-D, 2-D, and 3-D advection equations. In this paper, we extend the technique of MIEELDLD to the field of computational aeroacoustics and have been able to construct high-order methods with Low Dispersion and Low Dissipation properties which approximate the 1-D linear advection equation. Modifications to the spatial discretization schemes designed by Tam and Webb (1993), Lockard et al. (1995), Zingg et al. (1996), Zhuang and Chen (2002), and Bogey and Bailly (2004) have been obtained, and also a modification to the temporal scheme developed by Tam et al. (1993) has been obtained. These novel methods obtained using MIEELDLD have in general better dispersive properties as compared to the existing optimised methods.

Time-Splitting Procedures for the Numerical Solution of the 2D Advection-Diffusion Equation

We perform a spectral analysis of the dispersive and dissipative properties of two time-splitting procedures, namely, locally one-dimensional (LOD) Lax-Wendroff and LOD (1, 5) [9] for the numerical solution of the 2D advection-diffusion equation. We solve a 2D numerical experiment described by an advection-diffusion partial differential equation with specified initial and boundary conditions for which the exact solution is known. Some errors are computed, namely, the error rate with respect to the norm, dispersion and dissipation errors. Lastly, an optimization technique is implemented to find the optimal value of temporal step size that minimizes the dispersion error for both schemes when the spatial step is chosen as 0.025, and this is validated by numerical experiments.

Analysis of the unconditionally positive finite difference scheme for advection-diffusion-reaction equations with different regimes

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.

Comparison of some finite difference methods for the Black-Scholes equation

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.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.

A computational study of some numerical schemes for a test case with steep boundary layers

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.