General methods are often not adequate for accurate resolution of steep gradient phenomena; they usually introduce non-physical effects such as smearing of the solution or spurious oscillations. Since publication of Godunov's order barrier theorem, which proved that linear methods cannot provide non-oscillatory solutions higher than first order (, ), these difficulties have attracted much attention and a number of techniques have been developed that largely overcome these problems.
General methods are often inadequate for accurately resolving steep gradient phenomena; they typically introduce unwanted effects like smearing or spurious oscillations. Since Godunov's order barrier theorem, which proved that linear methods cannot provide non-oscillatory solutions higher than the first order, these difficulties have attracted much attention and a number of techniques have been developed to overcome them.