hi

when modelling 1D wave eqn. the dissipation error is due to artificial viscosity(For first order schemes).Can anybody explain how this artificial viscosity came into picture as there is no viscosity term either in the model eqn or modified eqn.

Model eqn:du_dt=-a du_dx

Modified eqn:du_dt=-a du_dx+ [(a*Delta x/2)*(1-c)*(d^2u_dx^2)].

where "a" is speed of sound and "C" is courant number.

As per text book the second term in the model eqn that is [(a*Delta x/2)*(1-c)] is called artificial viscosity. How this will become a viscous term ? Any help in this regard is highly appreciated.

rvndr

Artificial viscosity (= parabolic diffusive character) is often added to a numerical method to stabilize it and obtain non-oscillating solutions. This is achieved by adding a (negative = diffusive, as opposed to anti-diffusive) second order spatial derivative. To see how this is diffusive: treat the numerical solution as Fourier modes, ~ A_k * exp(i*k . x), and look at the eigenvalues of these operators,

d/dx = i*k (purely imaginary, so dispersive)

d^2/dx^2 = -k^2 (real and negative, so diffusive)

Is this of any help?