I am trying to solve Ideal MHD equation using semi discrete methods, ENO spatial reconstructions and TVD RK time stepping. I am getting different steady state solutions with different temporal order. Is it correct?
If the semi-discrete model has a unique steady state and no limit cycles, then all convergent methods should converge to the same steady state. If the problem has multiple steady states (hysteresis) or stable limit cycles, then any change to the method (RK method, initial time step, and even non-associativity of floating point when summing stages) can change the computed steady state. Since any convergent method approximates the constant solution correctly, the differences are in how you got there.