In order to apply these methods to a system , A should be Symmetric Positive definite For such cases, function f is defined such that, Then the gradient of f Hence, if f is minimised, then and it will make sure tat . It can be seen too in the other way around. Steepest descent …

Iterative methods: Jacobi and Gauss Seidel methods

Generalised analysis Solution Step 1: Step 2: (Jacobi) Step 2: (Gauss Siedal) In a general way, Jacobi Gauss Siedal: Convergence and spectral radius Expression for error represents the exact solution. Now, taking (2) – (3) gives the error(). . Relating error with eigen value If are the eigen values of M and are the corresponding …

Direct solvers

Direct solvers of system of algebraic equations. Gauss elimination Step 1:Making the first column zero Step 2:Making the second column elements (entries below diagonal) zero In general, kth column : Forward elimination: There are three loops in the forward elimination. Hence the order of computation is Backward substitution Algorithm for backward substitution: Here, the order …

Basic time integration schemes

Consider the unsteady one dimensional heat conduction without any source term. We shall integrate LHS and RHS before discretisation. In LHS, the time integration carries out first. notating Now, in order to carryout the space integration, we need to have an assumption for variation of T with respect to time t. Assuming the “piecewise linear” …

SIMPLE scheme (1/3): Theory

as I learned from the awesome book Derivation of SIMPLE scheme for 1D uniform grid A steady momentum equation may be wrote as follows. If it is one dimensional, the momentum and continuity equations are as follows (2) and (3): Representing pressure and velocity on the cell centre may lead to so called checker board …

High Resolution schemes

HR scheme = Higher Order Scheme + a Convection Boundedness Criterion HR scheme can be obtained by either of the following method Smoothening an unbounded Higher Order scheme with diffusive flux. Enhancing a first order upwind scheme with a limited anti-diffusive flux (TVD schemes). Two frameworks used for convection boundedness are Normalized Variable Formulation (NVF) …