Efficient exponential Runge–Kutta methods of high order: construction and implementation

Exponential Runge–Kutta methods have shown to be competitive for the time integration of stiff semilinear parabolic PDEs. The current construction stiffly accurate exponential methods, however, relies on a convergence result that requires weakening many order conditions, resulting in schemes whose stages must implemented sequential way. In this work, after showing stronger result, we are able derive two new families fourth- and fifth-order which, contrast existing multiple independent one another share same format, thereby allowing them parallel or simultaneously, making behave like using with much less stages. Moreover, all their involve only linear combination product $$\varphi $$ -functions (using argument) vectors. Overall, these features make more efficient implement when compared orders. Numerical experiments one-dimensional problem, nonlinear Schrödinger equation, two-dimensional Gray–Scott model given confirm accuracy efficiency newly constructed methods.