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Exact and Approximation Riemann Solutions of Nonlinear Hyperbolic Conservation Laws

Thesis Info

Access Option

External Link

Author

Rehman, Assad

Program

PhD

Institute

COMSATS University Islamabad

City

Islamabad

Province

Islamabad.

Country

Pakistan

Thesis Completing Year

2018

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/11175/1/Assad%20Complete%20thesis.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726149295

Similar


This thesis project focuses on the numerical solutions of selected nonlinear hyperbolic sys tems of partial differential equations (PDEs) describing incompressible and compressible flows. Such type of PDEs are used to simulate various flows in science and engineering. The underlying physics of such systems of PDEs is very complex and some mathematical and computational issues are associated with them. For instance, they may contain non conservative terms or may be weakly hyperbolic. The strong nonlinearity of the systems could generate sharp fronts in the solutions in a finite time interval, even for smooth initial data. Moreover, accurate discretization of the non-conservative terms is a challenge task for the numerical solution techniques. In the presence of non-conservative terms, well balancing, positivity preservation and capturing of steady states demand special attention during the application of a numerical algorithm. In this thesis project, we develop exact Riemann solvers for the one-dimensional Ripa model, containing shallow water equations that incorporate horizontal temperature gradients and considering both flat and non flat bottom topographies. Such Riemann solvers are helpful for understanding the behavior of solutions, as these solutions contain fundamental physical and mathematical characters of the set of conservation laws. Such solvers are also very helpful for evaluating performance of the numerical schemes for more complex models. Afterwards, third order well-balanced finite volume weighted essentially non-oscillatory (FV WENO) schemes are applied to solve the same model equations in one and two space dimensions and a Runge-Kutta discontin uous Galerkin (RKDG) finite element method is applied to solve this model in one space dimension. In the case of compressible fluid flow models, an upwind conservation element and solution element (CE/SE) method and third order finite volume WENO schemes are applied to solve the dusty gas and two-phase flow models. The suggested numerical schemes are able to tackle the above mentioned associated difficulties in a more efficient manner. The accuracy and order of convergence of the proposed numerical schemes are analyzed qualitatively and quantitatively. A number of numerical test problems are considered and results of the suggested numerical schemes are compared with the derived exact Riemann solutions, results available in the literature, and with the results of a high resolution central upwind (CUP) scheme.
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حدود کی تعداد

حدود کی تعداد
جرائم حدود کی تعداد کے سلسلے میں اہل علم میں اختلاف پایا جاتا ہے۔ ابن حجر عسقلانی کے نزدیک
" جَمْعُ حَدٍّ وَالْمَذْكُورُ فِيهِ هُنَا حَدُّ الزِّنَا وَالْخَمْرِ وَالسَّرِقَةِ وَقَدْ حَصَرَ بَعْضُ الْعُلَمَاءِ مَا قِيلَ بِوُجُوبِ الْحَدِّ بِهِ فِي سَبْعَةَ عَشَرَ شَيْئًا فَمِنَ الْمُتَّفَقِ عَلَيْهِ الرِّدَّةُ وَالْحِرَابَةُ مَا لَمْ يَتُبْ قَبْلَ الْقُدْرَةِ وَالزِّنَا وَالْقَذْفُ بِهِ وَشُرْبُ الْخَمْرِ سَوَاءٌ أَسَكِرَ أَمْ لَا وَالسَّرِقَةُ وَمِنَ الْمُخْتَلَفِ فِيهِ جَحْدُ الْعَارِيَةِ وَشُرْبُ مَا يُسْكِرُ كَثِيرُهُ مِنْ غَيْرِ الْخَمْرِ وَالْقَذْفُ بِغَيْرِ الزِّنَا وَالتَّعْرِيضُ بِالْقَذْفِ وَاللِّوَاطُ وَلَوْ بِمَنْ يَحِلُّ لَهُ نِكَاحُهَا وَإِتْيَانُ الْبَهِيمَةِ وَالسِّحَاقُ وَتَمْكِينُ الْمَرْأَةِ الْقِرْدَ وَغَيْرَهُ مِنَ الدَّوَابِّ مِنْ وَطْئِهَا وَالسِّحْرُ وَترك الصَّلَاة تكاسلا وَالْفطر فِي رَمَضَان وَهَذَا كُلُّهُ خَارِجٌ عَمَّا تُشْرَعُ فِيهِ الْمُقَاتَلَةُ كَمَا لَوْ تَرَكَ قَوْمٌ الزَّكَاةَ وَنَصَبُوا لِذَلِكَ الْحَرْبَ "64
"ابن حجر عسقلانی نے سترہ جرائم کو حدود میں شامل کیا ہے اور گیارہ جرائم کے متعلق اتفاق ظاہر کیا ہے کہ یہ حدود میں شامل ہیں جو کہ مند رجہ ذیل ہیں ۔ زنا ، قذف ، سرقہ ، بغاوت، شراب نوشی، ارتداد ، حرابہ، ترک صلوٰۃ ، ترک صوم، سحر اور وطی بہائم۔ "
علامہ کاسانی ؒکے مطابق حدود کی تعداد پانچ ہے، جو کہ مندرجہ ذیل ہیں:
"الْحُدُودُ خَمْسَةُ أَنْوَاعٍ حَدُّ السَّرِقَةِ وَحَدُّ الزِّنَا وَحَدُّ الشُّرْبِ وَحَدُّ السُّكْرِ وَحَدُّ الْقَذْفِ۔ "65
ابن قدامہ ؒ حدود کی تعداد سات کا ذکر کرتے ہوئے لکھتے ہیں
"الْحُدُودُ سبعۃأَنْوَاعٍ حَدُّ السَّرِقَةِ وَحَدُّ الزِّنَا وَحَدُّ الشُّرْبِ وَحَدُّ الْقَذْفِوَحَدُّ الحرابۃ وحد الردۃ وحد بغی۔ "66
عبدالقادر عودہ شہید ؒکے بقول حدود کی تعداد سات ہے ، جو کہ یہ ہیں:
"جرائم الحدود معینۃ ومحددۃ العدد وھی سبع جرائم حَدُّ الزِّنَا وَحَدُّ السُّكْرِ وَحَدُّ الْقَذْفِ حَدُّ السَّرِقَةِوَحَدُّ الحرابۃ وحد ارتداد وحد بغی "67
جسٹس تنزیل الرحمن کے مطابق حدود اللہ کی تعداد چھ ہے
"حَدُّ الزِّنَا وَحَدُّ الْقَذْفِ وَحَدُّ الشُّرْبِ حَدُّ السَّرِقَةِ وَحَدُّ الحرابۃوحد الردۃ۔...

تفسیرِ ماجدی کے منتخب آیات کی روشنی میں یہودیت کے متعلق مولانا عبد الماجد دریابادی کی آراء کا تحقیقی و تجزیاتی مطالعہ

There is no doubt in the fact that Judaism is the oldest Abrahamic religion among all. Judaism is not only the oldest religion, but it provides a foundation for the coming two Abrahamic religions as well i.e. Christianity and Islam. There have already been rigorous discussions in the existing literature regarding the beliefs and history of Judaism. Scholars have also shed light on the beliefs of various sects of Jews. ‘Tafseeri-Majdi’ is one such contribution to the literature. The present study focuses on the work of Majdi and discusses various famous theories, stories, and personalities presented in his ‘Tafseer’. The present study attempts to shed light on the life of Abdul Majid Daryabadi, the methodology adopted in his ‘Tafseer’, and analysis of his approaches and discussions about Judaism.

Semi-Automated Landslide Detection and Susceptibility Modelling Using Geospatial Tools in Northern Pakistan

Landslides are frequent hazard in the mountainous region of Pakistan with significant threat to surrounding infrastructure and communities. Landslide inventories and susceptibility maps are prerequisite for landslide hazard mitigation; however, these are rarely available for most of the mountainous areas in northern Pakistan. Traditionally, landslide inventory is manually developed through visual interpretation of remote sensing image and aerial photographs. Recently, Object-Based Image Analysis (OBIA) image classification technique is effectively applied to high resolution remote sensing data for rapid and precise landslide detection and temporal change analysis. However, due to unavailability of timely and optimal remote sensing data, OBIA methods are rarely applied for quick post-disaster landslide inventory and susceptibility assessment. Aim of this study is to utilize manual and semi-automated techniques for landslide inventories and susceptibility assessment. The selected study areas include Hunza-Nagar valley of Gilgit Baltistan and Muzaffarabad region of Azad Kashmir in northern Pakistan.For Hunza Nagar valley, landslide inventory was developed using visual interpretation of the SPOT-5 multi-spectral data in 3D environment. The acquired landslide inventory is correlated with seven landslide causative factors through Weight of Evidence and Frequency Ratio techniques, to developed landside susceptibility map for the study area. The produced landslides susceptibility maps are validated by the success rate and area under curves criteria. The prediction powers of the statistical models are also validated with the prediction rate curve. It is observed that Weight of Evidence modeling is suited for landslide susceptibility mapping in the study area. The developed landslide inventory and susceptibility map can be used for landslide disaster mitigation strategies. However, it was observed that manual development of landslide inventory over a large area is laborious and time consuming.OBIA technique is used for rapid and precise detection of landslides in the Muzaffarabad and Balakot surrounding regions in northern Pakistan. Transferability and efficiency of the three existing OBIA landslide detection methods are assessed for landslide detection. OBIA technique is modified and applied to the SPOT-5, SPOT-6 and ALOS PALSAR DEM data for landslides detection in the Muzaffarabad and Balakot region. The SPOT-6 multi-spectral data with ALOS PALSAR DEM derivatives e.g. slope, aspect, hill-shade, vi elevation and streams are used for semi-automated landslide detection. The spectral, contextual, textural, spatial and morphological characteristics of landslide in remote sensing image are assessed for landslide identification. Moreover, a semi-automated method is developed to use the recently launched and freely available Sentinel-2 MSI data for near-real time landslide monitoring and temporal change detection. NDVI, terrain slope and mean brightness were found useful for landslide recognition in the area. The resultant semi-automated landslide inventories are validated through confusion matrix. The temporal change analysis shows that landslides area and numbers are increased from the year 2016 to 2018 in the study area. The developed semi-automated method using the Sentinel-2 time series data depicts that the Sentenil-2 data could be used for rapid post-disaster landslide detection, monitoring and spatio-temporal change analysis in the landslide prone areas over a regional scale. OBIA based landslide inventory for the year 2018 used for landslide susceptibility assessment in the study area. The proposed OBIA method can help to develop landslide inventory relatively quickly, and hereafter has the capability to be used for landslide susceptibility, hazard, vulnerability and risk assessment. Moreover, the results from this study can help in disaster risk mitigation and quick inventory and susceptibility assessment in the aftermath of a landslide triggering events like earthquake or rainfall.