High-order isogeometric discretizations produce a small number of non-physical outlier modes at the top of the discrete spectrum. These modes can pollute the numerical solution in wave propagation and structural dynamics. Most existing remedies modify the spatial discretization through outlier-free bases, mass lumping, subspace projection, and related constructions. We introduce MIISO (Modal Integrators for Isogeometric Stabilization of Outliers), a family of exponential Rosenbrock-Krylov methods of orders two through six that suppress the outliers in the time integrator and leave the spatial discretization unchanged. We construct a non-holomorphic amplification map by blending a Padé approximation that preserves physical oscillatory modes with a Pade approximation that damps high frequencies, with the blend placed at the known spectral location of the outliers. For orders three and higher the damping approximant is repaired so that the method is unconditionally A-stable. Physical modes receive no numerical dissipation and the outliers are driven to zero. The outlier handling and the method steps are fully matrix-free, and the method extends to nonlinear problems. Numerical experiments recover the predicted orders, match spatial outlier removal without changing the discretization, show lower cost than generalized-alpha at matched accuracy and in scaling, and demonstrate the method on nonlinear and three-dimensional problems.

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