High-order advection schemes are investigated in linear and nonlinear advection. Numerical tests and theoretical analyses indicate that the performance level of semi-Lagrangian advection schemes using cubic spline interpolation is between those using quintic and seventh-order Lagrange interpolations in uniform resolution. However, higher-order Lagrange interpolations yield larger dispersion in the region of variable resolution in the rotational flow test. The degrading phase preservation can be applied in combination with higher-order Lagrange interpolations to suppress the dispersion of negative values. In variable resolution, simple Eulerian formulations generally have insufficient accuracy and stringent stability constraints, but semi-Lagrangian schemes do not. Based on the linear and nonlinear advection tests, it was suggested that variable-resolution mesoscale models may employ cubic spline or cubic B-spline for horizontal advection and cubic Lagrange interpolation for vertical advection without recourse to quintic and higher-order Lagrange interpolations.