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

A note on the shock-capturing properties of some explicit and implicit schemes for solving the 1-D linear advection equation

The technique of Minimized Integrated Exponential Error for Low Dispersion and Low Dissipation, (MIEELDLD) was introduced in Appadu and Dauhoo (2011), Appadu and Dauhoo (2009) and extensive work on this technique is reported further in Appadu (In Press), Appadu (2012). The technique enables us to assess the shock-capturing properties of numerical methods. It also allows us to nd suitable values for parameters present in numerical methods in order to optimise their dissipative and dispersive properties (Appadu 2012). This technique basically makes use of a physical quantity called the Integrated Ex- ponential Error for Low Dispersion and Low Dissipation, IEELDLD. In this work, we obtain the IEELDLD for some explicit, quasi-implicit and implicit meth- ods. We use MIEELDLD to obtain an explicit scheme with more effective shock-capturing properties than Gadd and Carpenter's numerical schemes. Also, an implicit method is con- structed which is almost similar to the one derived by Dehghan (2005) and which has also better shock-capturing properties as compared to the Crank-Nicolson method.This work was supported by a postdoctoral fellowship at the University of Cape Town,funded through the South African research Chair in Computational Mechanics.http://www.journals.elsevier.com/journal-of-applied-mathematics-and-mechanicshb2016Mathematics and Applied Mathematic

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 ID 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 ID linear advection and ID Burger’s equation at some different cfl numbers and show that the optimal cfl does indeed cause less dispersion, less dissipation, and lower L1 errors. Lastly, we test numerically the order of convergence of the WENO3 scheme.The Research Development Programme (RDP) of the University of Pretoriahttp://www.hindawi.com/journals/jam/am201

Some optimised schemes for 1D Korteweg-de-Vries equation

Two new explicit finite difference schemes for the solution of the one-dimensional Korteweg-de-Vries equation are proposed. This equation describes the character of a wave generated by an incompressible fluid. We analyse the spectral properties of our schemes against two existing schemes proposed by Zabusky and Kruskal (1965) and Wang et al. (2008). An optimisation technique based on minimisation of the dispersion error is implemented to compute the optimal value of the spatial step size at a given value of the temporal step size and this is validated by some numerical experiments. The performance of the four methods are compared in regard to dispersive and dissipative errors and their ability to conserve mass, momentum and energy by using two numerical experiments which involve solitons.South African DST/NRF SARChI Chair on Mathematical Models and Methods in Bioengineering and Biosciences (M3B2) of the University of Pretoria and the National Research Foundation of South Africa [95864, 93476].http://www.inderscience.com/jhome.php?jcode=PCFD2017-12-21hj2017Mathematics and Applied Mathematic

The technique of MIEELDLD as a measure of the shock-capturing property of numerical methods for hyperbolic conservation laws

In this paper, we use some numerical methods namely Lax-Wendroff (LW), two-step Lax- Friedrichs (LF), two variants of composite methods made up of Lax-Wendroff and the twostep Lax-Friedrichs and Fromm’s scheme to solve a 1D linear advection and 1D diffusionless Burger’s equation, at some values of the Courant number. We then use two optimisation techniques based on both dispersion and dissipation and two optimisation techniques based on only dispersion and obtain the variation of the integrated errors vs the CFL number. It is seen that out of the five techniques, only one is a good measure of the shock-capturing of property of numerical methods.Research Development Programme of the University of Pretoriahttp://www.inderscience.com/jhome.php?jcode=PCFD2015-10-31hb201

Comparative gene expression profiling of ADAMs, MMPs, TIMPs, EMMPRIN, EGF-R and VEGFA in low grade meningioma

MMPs (matrix metalloproteinases), ADAMs (a disintegrin and metalloproteinase) and TIMPs (tissue inhibitors of metalloproteinases) are implicated in invasion and angiogenesis: both are tissue remodeling processes involving regulated proteolysis of the extracellular matrix, growth factors and their receptors. The expression of these three groups and their correlations with clinical behaviour has been reported in gliomas but a similar comprehensive study in meningiomas is lacking. In the present study, we aimed to evaluate the patterns of expression of 23 MMPs, 4 TIMPs, 8 ADAMs, selective growth factors and their receptors in 17 benign meningiomas using a quantitative real-time polymerase chain reaction (qPCR). Results indicated very high gene expression of 13 proteases, inhibitors and growth factors studied: MMP2 and MMP14, TIMP-1, -2 and -3, ADAM9, 10, 12, 15 and 17, EGF-R, EMMPRIN and VEGF-A, in almost every meningioma. Expression pattern analysis showed several positive correlations between MMPs, ADAMs, TIMPs and growth factors. Furthermore, our findings suggest that expression of MMP14, ADAM9, 10, 12, 15 and 17, TIMP-2, EGF-R and EMMPRIN reflects histological subtype of meningioma such that fibroblastic subtype had the highest mRNA expression, transitional subtype was intermediate and meningothelial type had the lowest expression. In conclusion, this is the first comprehensive study characterizing gene expression of ADAMs in meningiomas. These neoplasms, although by histological definition benign, have invasive potential. Taken together, the selected elevated gene expression pattern may serve to identify targets for therapeutic intervention or indicators of biological progression and recurrence

A computational study of three numerical methods for some advection-diffusion problems

Three numerical methods have been used to solve two problems described by advection-diffusion equations with specified initial and boundary conditions. The methods used are the third order upwind scheme [4], fourth order upwind scheme [4] and Non-Standard Finite Difference scheme (NSFD) [9]. We considered two test problems. The first test problem has steep boundary layers near x = 1 and this is challenging problem as many schemes are plagued by non-physical oscillation near steep boundaries [15]. Many methods suffer from computational noise when modelling the second test problem especially when the coefficient of diffusivity is very small for instance 0.01. We compute some errors, namely L2 and L1 errors, dissipation and dispersion errors, total variation and the total mean square error for both problems and compare the computational time when the codes are run on a matlab platform. We then use the optimization technique devised by Appadu [1] to find the optimal value of the time step at a given value of the spatial step which minimizes the dispersion error and this is validated by some numerical experiments.Research Development Programme of the University of Pretoria and the DST/NRF SARChI Chair in Mathematical Models and Methods in Bioengineering and Biosciences. Incentive fund N00 401 Project 85796, University of Pretoria, African Institute for Mathematical Sciences (AIMS)-South Africa and Aksum University (Ethiopia).http://www.elsevier.com/locate/amc2017-01-31hb201

A priori analysis of multilevel finite volume approximation of 1D convective Cahn–Hilliard equation

In this work, we analyze four finite volume methods for the nonlinear convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The methods used are: implicit one-level, explicit one-level, implicit multilevel and explicit multilevel finite volume methods. The existence and uniqueness of solution, convergence and stability of the finite volume solutions are proved. We compute L2- error and rate of convergence for all methods. We then compare the multilevel methods with the one-level methods by means of stability and CPU time. It is shown that the multilevel finite volume method is faster than the one-level method.A.R. Appadu is grateful to the DST/NRF SARChI Chair in Mathematical Models and Methods in Bioengineering and Biosciences and the National Research Foundation of South Africa Grant No. 95864 for funding. J. K. Djoko is funded through the incentive fund N00 401 Project 85796. H. H. Gidey is grateful to the University of Pretoria, African Institute for Mathematical Sciences (AIMS)-South Africa and Aksum University (Ethiopia) for their financial support for his Ph.D. studies.https://link.springer.com/journal/133702018-12-01hj2017Mathematics and Applied Mathematic

Computational study of three numerical methods for some linear and nonlinear advection-diffusion-reaction problems

In this paper, we use three existing schemes namely, Upwind Forward Euler, Non-Standard Finite Difference (NSFD) and Unconditionally Positive Finite Difference (UPFD) schemes to solve two numerical experiments described by a linear and a non-linear advection-diffusion-reaction equation with constant coefficients. These equations model exponential travelling waves and biofilm growth on a medical implant respectively. We study the exact and numerical dissipative and dispersive properties of the three schemes for both problems. Moreover, L1 error, dispersion and dissipation errors, at some values of temporal and spatial step sizes have been computed for the three schemes for both problems.DST/NRF SARChI chair in Mathematical Models and Methods in Bioengineering and BioSciences of the University of Pretoria.http://www.inderscience.com/jhome.php?jcode=PCFD2017-08-31hb2017Mathematics and Applied Mathematic

Analysis of multilevel finite volume approximation of 2D convective Cahn–Hilliard equation

In this work, four finite volume methods have been constructed to solve the 2D convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. We prove existence and uniqueness of solutions. The stability and convergence analysis of the numerical methods have been discussed thoroughly. The nonlinear terms are approximated by a linear expression based on Mickens’ rule (Mickens, Nonstandard finite difference models of differential equations. World Scientific, Singapore, 1994) of nonlocal approximations of nonlinear terms. Numerical experiments for a test problem have been carried out to test all methods.http://link.springer.com/journal/131602018-04-30hb2017Mathematics and Applied Mathematic