High-order low-dissipation low-dispersion diagonally implicit Runge-Kutta schemes are analyzed and introduced, based on the optimization of amplification and phase errors for wave propagation. Various optimized schemes can be obtained. The new scheme shows negligible dissipation. It is illustrated mathematically and numerically that the new scheme preserves fourth-order accuracy, while the recently developed diagonally implicit Runge-Kutta scheme does not. The numerical applications contain the advection equation with and without a stiff nonlinear source term and an oscillatory test. The new scheme is A-stable as desired for the solution of stiff problems.