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\contentsline {section}{\numberline {1}Introduction}{3}% 
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\contentsline {subsection}{\numberline {1.1}From reality to models. Computational prediction}{3}% 
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\contentsline {subsubsection}{\numberline {1.1.1}Ordinary Differential Equations (ODEs)}{3}% 
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\contentsline {paragraph}{Newton's second law:~}{3}% 
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\contentsline {paragraph}{Case of pure gravity:~}{3}% 
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\contentsline {paragraph}{String at rest between two walls:~}{4}% 
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\contentsline {subsubsection}{\numberline {1.1.2}Partial Differential Equations (PDEs)}{5}% 
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\contentsline {paragraph}{A soap membrane at rest}{5}% 
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\contentsline {paragraph}{Boundary conditions:~}{6}% 
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\contentsline {paragraph}{Circular bubble:~}{6}% 
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\contentsline {subsubsection}{\numberline {1.1.3}Computational prediction}{7}% 
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\contentsline {paragraph}{Approximate problem:~}{7}% 
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\contentsline {paragraph}{The algorithm cost is technically reachable:~}{8}% 
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\contentsline {paragraph}{The continuous problem is well-posed:~}{8}% 
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\contentsline {paragraph}{The numerical method converges toward the exact solution:~}{8}% 
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\contentsline {paragraph}{The numerical method is stable:~}{8}% 
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\contentsline {subsection}{\numberline {1.2}PDE Classification}{10}% 
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\contentsline {subsection}{\numberline {1.3}On the necessity to weaken the notion of derivative}{10}% 
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\contentsline {subsubsection}{\numberline {1.3.1}Linear scalar transport equation}{10}% 
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\contentsline {subsubsection}{\numberline {1.3.2}Non-Linear Burgers' equation}{11}% 
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\contentsline {section}{\numberline {2}Some tools for the theoretical analysis of PDEs}{13}% 
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\contentsline {subsection}{\numberline {2.1}Algebraical and topological duals}{13}% 
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\contentsline {subsection}{\numberline {2.2}Some functional spaces}{13}% 
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\contentsline {subsection}{\numberline {2.3}Weak derivatives}{15}% 
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\contentsline {subsubsection}{\numberline {2.3.1}Definition}{15}% 
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\contentsline {subsubsection}{\numberline {2.3.2}Examples}{16}% 
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\contentsline {subsection}{\numberline {2.4}Weak formulation of a PDE}{17}% 
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\contentsline {subsection}{\numberline {2.5}Sobolev spaces}{18}% 
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\contentsline {subsection}{\numberline {2.6}Lax-Milgram theorem and well-posedness}{19}% 
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\contentsline {subsection}{\numberline {2.7}Finite element approximation}{19}% 
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\contentsline {section}{\numberline {3}Numerical discretization of hyperbolic conservation laws}{22}% 
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\contentsline {subsection}{\numberline {3.1}What is a hyperbolic conservation law?}{22}% 
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\contentsline {subsubsection}{\numberline {3.1.1}Description}{22}% 
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\contentsline {paragraph}{Examples of hyperbolic conservation laws}{24}% 
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\contentsline {subsubsection}{\numberline {3.1.2}Characteristic curves}{24}% 
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\contentsline {paragraph}{Examples: }{25}% 
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\contentsline {subsubsection}{\numberline {3.1.3}Weak formulation}{25}% 
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\contentsline {subsubsection}{\numberline {3.1.4}Rankine-Hugoniot condition}{26}% 
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\contentsline {paragraph}{Example: Burgers' flux, $f(u) = u^2/2$.}{27}% 
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\contentsline {subsubsection}{\numberline {3.1.5}Non-Uniqueness of the weak solution}{27}% 
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\contentsline {subsubsection}{\numberline {3.1.6}Entropy solution}{28}% 
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\contentsline {subsubsection}{\numberline {3.1.7}Maximum Principle}{30}% 
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\contentsline {subsubsection}{\numberline {3.1.8}Boundary Conditions}{31}% 
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\contentsline {subsection}{\numberline {3.2}Numerical Methods for hyperbolic conservation Laws}{31}% 
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\contentsline {subsubsection}{\numberline {3.2.1}Finite Differences}{32}% 
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\contentsline {paragraph}{Explicit Schemes:~}{32}% 
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\contentsline {paragraph}{Von Neumann's Analysis:~}{33}% 
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\contentsline {paragraph}{Implicit schemes}{34}% 
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\contentsline {subsubsection}{\numberline {3.2.2}Stabilized Finite Elements}{34}% 
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\contentsline {paragraph}{Stabilization}{35}% 
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\contentsline {subsubsection}{\numberline {3.2.3}Finite Volume Methods}{36}% 
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\contentsline {paragraph}{Lax-Friedrichs Flux}{36}% 
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\contentsline {paragraph}{Lax-Wendroff Flux}{36}% 
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\contentsline {paragraph}{Mac-Cormack Method}{37}% 
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\contentsline {paragraph}{Godunov Method}{37}% 
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\contentsline {subsubsection}{\numberline {3.2.4}Discontinuous Galerkin Methods}{37}% 
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\contentsline {subsubsection}{\numberline {3.2.5}Residual Distribution Schemes}{37}% 
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\contentsline {subsubsection}{\numberline {3.2.6}Time Integration}{37}% 
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\contentsline {subsection}{\numberline {3.3}Positivity and convex constraints preservation}{37}% 
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\contentsline {section}{\numberline {4}Numerical treatment of some SDEs}{38}% 
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\contentsline {subsection}{\numberline {4.1}Random process and Brownian motion}{38}% 
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\contentsline {subsection}{\numberline {4.2}Stochastic integral}{40}% 
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\contentsline {subsection}{\numberline {4.3}Stochastic differential equations and PDEs}{41}% 
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\contentsline {subsection}{\numberline {4.4}Numerical treatment of stochastic differential equations}{42}% 
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\contentsline {section}{\numberline {5}Conclusion}{43}% 
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\contentsline {section}{\numberline {6}References}{44}% 
