Sea ice dynamics determine the drift and deformation of sea ice. Nonlinear physics, usually expressed in a viscous‐plastic rheology, makes the sea ice momentum equations notoriously difficult to solve. At increasing sea ice model resolution the nonlinearities become stronger as linear kinematic features (leads) appear in the solutions. Even the standard elastic‐viscous‐plastic (EVP) solver for sea ice dynamics, which was introduced for computational efficiency, becomes computationally very expensive, when accurate solutions are required, because the numerical stability requires very short, and hence more, subcycling time steps at high resolution. Simple modifications to the EVP solver have been shown to remove the influence of the number of subcycles on the numerical stability. At low resolution appropriate solutions can be obtained with only partial convergence based on a significantly reduced number of subcycles as long as the numerical procedure is kept stable. This previous result is extended to high resolution where linear kinematic features start to appear. The computational cost can be strongly reduced in Arctic Ocean simulations with a grid spacing of 4.5 km by using modified and adaptive EVP versions because fewer subcycles are required to simulate sea ice fields with the same characteristics as with the standard EVP.