Browsing by Author "CIIT/FA10-MSMATH-011/LHR"

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    Dispersion Free Finite Difference Scheme for Time Harmonic Wave Equation
    (COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2012) CIIT/FA10-MSMATH-011/LHR; Muhammad Sarmad Arshad 05/LHR; Dr. Hafiz Abdul Wajid; FA10
    Dispersion Free Finite Difference Scheme for Time Harmonic Wave Equation In this work, we construct novel central finite difference schemes for solving the Helmholtz equation by following the idea of Nehrbass and Yau Shu in one and two dimensions. We present an alternative construction by which not only central f inite difference schemes but all types of schemes (forward, backward) can be made exact for Helmholtz equation. We give explicit expressions for the coefficient of the central node in the standard central finite difference scheme of any order which makes the novel scheme dispersion free in one dimension and optimal in two dimensions. Such schemes add no cost for the implementation and one needs not to write brand new code instead just need to change the central node coefficient with the new one presented in this work to have highly accurate results. Comparison of numerical solutions obtained using novel schemes with standard schemes for Helmholtz equation in single and higher dimensions are given.
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    Optimal Finite Difference Schemes for Wave Equation
    (COMSATS University Islamabad, Lahore Campus Library Information Services, CUI Lahore, 2012) Hifza Iqbal; CIIT/FA10-MSMATH-011/LHR; Master of Science in Mathematics; Dr. Hafiz Abdul Wajid
    In this research work the novel optimal central finite difference schemes for solving the transient wave equation, are developed and analyzed. The novel schemes (a) give exact value at the nodes of the grid, meaning that an exact solution is possible for the transient wave equation; (b) the key advantage is that they have the simple structure as that of the standard central finite difference schemes, so we just need to make few amendments in the standard codes, instead of writing the new codes altogether; (c) above all, the beauty lies in the fact that they do not require fine mesh size and works for any wave number, specially for high wave number. Finally, in order to display the actual power of the novel schemes, numerical computations and graphical representations for two model problems are added, by using both novel and standard central finite difference schemes, which support our theoretical results, by showing the superiority of the novel schemes compared to the standard schemes