jats:titleABSTRACT</jats:title> jats:pThe non-linear behaviour of low-viscosity warped discs is poorly understood. We verified a non-linear bending-wave theory, in which fluid columns undergo affine transformations, with direct 3D hydrodynamical simulations. We employed a second-order Godunov-type scheme, meshless finite mass (MFM), and also the smoothed particle hydrodynamics (SPH) method, with up to 128 million particles. For moderate non-linearity, MFM maintains well the steady non-linear warp predicted by the affine model for a tilted inviscid disc around a central object with a quadrupole moment. However, numerical dissipation in SPH is so severe that even a low-amplitude non-linear warp degrades at a resolution where MFM performs well. A low-amplitude arbitrary warp tends to evolve towards a non-linear steady state. However, no such state exists in our thin disc with an angular semithickness H/R = 0.02 when the outer tilt angle is beyond about 14°. The warp breaks tenuously and reconnects in adiabatic simulations, or breaks into distinct annuli in isothermal simulations. The breaking radius lies close to the location with the most extreme non-linear deformation. Parametric instability is captured only in our highest resolution simulation, leading to ring structures that may serve as incubators for planets around binaries.</jats:p>