Question

Numerical Methods for Conservation Laws, Randall J. LeVeque, Ed. 2.

1. Consider the linear equations, Use central difference for the spatial discretization. Derive the stability condition for the schemes with the followinor temporal discretization

question/ do 2nd order explicit Runge-Kutta

hint

Answer 1

GOMR Nicolson differencing scheme: partial differential T/partial differential t = k partial differential^2 T/partial differential x^2 partial differential T/ partial differential t = T_i, j + 1 - T_i, j/delta t partial differential^2 T/partial differential x^2 = 1/2 [T_(I + 1, J) - 2T_i, j + T_i - 1, j/(delta x)^2 + T_i + 1, J + 1 - 2T_i, J = 1 + T_i - 1, J + 1/delta x^2] T_i, j + 1 - T_i, j/delta t = k/2 [1/2 [T_(I + 1, J) - 2T_i, j + T_i - 1, j/delta x^2 + T_i + 1, J + 1 - 2T_i, J = 1 + T_i - 1, J + 1/delta x^2] T_i, J + 1 - T_ij = k delta t/2 delta x^2 [T_i + 1, j - 2T_iji + T_i - 1, j + T_i + 1, j + 1 - 2T_i, j + 1 + T_i - 1, j + 1] 2 (T_i, J + 1 - T_ij) = lambda [T_i + 1, j - 2T_ij + T_i - 1, j + T_i + 1,