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This course studies basic methods for the numerical solution of partial differential
equations. Emphasis is placed on regarding the discretized algebraic equations
as discrete models of the system being studied. Example applications are drawn
from: pollutant transport, computational fluid dynamics, computational finance,
simulation of optical transmission lines.
The current course textbook is: Computational Differential Equations, by E.
Eriksson, D. Estep, P. Hansbo and C. Johnson, 1996, Cambridge.
Normally available in Winter.
Description of Types of Equations (3 hrs)
Informal classification of hyberbolic, parabolic and elliptic equations. Speed of information propagation. Impact of equation type on numerical algorithm.
Finite Volume Methods (3 hrs)
Finite volume methods, contrast with finite difference. Convergence, truncation error. Application to conservation laws.
Parabolic Equations (6 hrs)
Stability, explicit, implicit methods, monotone schemes. Non-linearities.
Elliptic Equations (9 hrs)
Finite element methods. Galerkin discretization. Relationship to finite volume methods. Assembly of stiffness matrix. Use of sparse matrix software, sparse matrix data structures. Numerical properties of M-matrices.
Hyperbolic Equations (6 hrs)
First order linear and non-linear hyperbolic equations. Discretizations of positive type, relation to TVD and monotone methods. Shocks and the E-condition, upstream weighting.
Equations in Non-conservative Form (3 hrs)
Examples of HJB type equations. Applications in optimal control and finance. Discretization methods.
Systems of Equations (6 hrs)
Discretization and solution of algebraic equations. Examples drawn from: Navier-Stokes, Euler equations, multi-phase subsurface flow, structural analysis.