Numerical methods 1 An Introduction to Numerical Methods For Weather Prediction by Mariano Hortal office 122

Numerical methods 2 Shallow water equations in 1 dimension advection adjustement diffusion

Numerical methods 3 Linearization u=U 0 +u’ h=H +h’ Basic + perturbation Substitute and drop products of perturbations

Numerical methods 4 Classification of PDE’s Boundary value problems Initial value problems D ΓDΓD D is an open domain Γ D its boundary L: differential operator φ: unknown function of x φ: unknown function of t t=0 t

Numerical methods 5 Initial and boundary value problems Eigenvalue problems Classification of PDE’s (II) φ: unknown eigenfunction λ: eigenvalue of operator L

Numerical methods 6 Linear second order partial differential equation Hyperbolic if b 2 -4ac>0 Parabolic if b 2 -4ac=0 (discriminant) Elliptic if b 2 -4ac<0

Numerical methods 7 Existence and uniqueness of solutions ; y(t 0 )=y 0 (initial value problem) -Does it have a solution? -Does it have only one solution? -Do we care? If it has one and only one solution it is called a well posed problem

Numerical methods 8 Picard’s Theorem Let andbe continuous in the rectangle then, the initial-value problem Has a unique solution y(t) on the interval Finding the solution (not analytical) Numerical methods (finite dimensions)

Numerical methods 9 Discretization Finite differences Spectral Finite elements Transform the continuous differential equation into a system of ordinary algebraic equations where the unknowns are the numbers f j

Numerical methods 10 Example with a boundary value problem Finite differences Spectral or finite elements in xєD in xєΓ (Dirichlet) approximate derivative analytical derivative Galerkin approach

Numerical methods 11 Minimizing the error Consider the problem: Where H is a linear operator in the space coordinate Approximate Minimizing:with respect to the unknowns gives Which is a particular case of the Galerkin procedure

Numerical methods 12 Further comments If the basis functions “e” are orthogonal, we have a system of decoupled equations –Spectral technique: The basis functions are orthogonal –Finite elements: The basis functions are locally non- zero Initial conditions: Which is a system of coupled equations in the unknowns If the basis functions are orthonormal, then which are m decoupled equations

Numerical methods 13 Convergence Consistency Stability Lax-Richtmeyer theorem Discretized equation > continuous equation discretization finer and finer The Lax-Richtmeyer theorem Discretized solution > continuous solution discretization finer and finer Discretized solution bounded If a discretization scheme is consistent and stable then it is convergent, and viceversa