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Astrophysical fluid dynamical problems rely on efficient numerical solution techniques for hyperbolic and parabolic terms. Efficient techniques are available for treating the hyperbolic terms. Parabolic terms, when present, can dominate the time for evaluating the solution, especially when large meshes are used. This stems from the fact that the explicit time‐step for parabolic terms is proportional to the square of the mesh size and can become unusually small when the mesh is large. Multigrid‐Newton–Krylov methods can help, but usually require a large number of iterations to converge. Super TimeStepping schemes are an interesting alternative, because they permit one to take very large overall time‐steps for the parabolic terms while using only a modest number of explicit time‐steps. Super TimeStepping schemes of the type used in astrophysics have, so far, been only first‐order accurate in time and prone to instabilities. In this paper, we present a Runge–Kutta method that is based on the recursion sequence for Legendre polynomials, called the RKL2 method. RKL2 is a time‐explicit method that permits us to treat non‐linear parabolic terms robustly and with large, second‐order accurate time‐steps. An s‐stage RKL2 scheme permits us to take a time‐step that is ∼s2 times larger than a single explicit, forward Euler time‐step for the parabolic operator. This permits an s‐fold gain in computational efficiency over explicit time‐step sub‐cycling. For modest values of ‘s’, the advantage can be substantial. The stability properties of the new schemes are explored and they are shown to be stable and positivity preserving for linear operators. We document the method as it is applied to the anisotropic thermal conduction operator for dilute, magnetized, astrophysical plasmas. Implementation‐related details are discussed. The RKL2 Super TimeStepping scheme has been implemented in the riemann code for computational astrophysics. We explain the method for picking an s‐stage RKL2 scheme for the parabolic terms and show how it can be integrated with a hyperbolic system solver. The method’s simplicity makes it very easy to retrofit the s‐stage RKL2 scheme to any problem with a parabolic part when a well‐formed spatial discretization is available. Several stringent test problems involving thermal conduction in astrophysical plasmas are presented and the method is shown to perform robustly and efficiently on all of them.

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A common attribute of capturing schemes used to find approximate solutions to the Euler equations is a sub-linear rate of convergence with respect to mesh resolution. Purely nonlinear jumps, such as shock waves produce a first-order convergence rate, but linearly degenerate discontinuous waves, where present, produce sub-linear convergence rates which eventually dominate the global rate of convergence. The classical explanation for this phenomenon investigates the behavior of the exact solution to the numerical method in combination with the finite error terms, often referred to as the modified equation. For a first-order method, the modified equation produces the hyperbolic evolution equation with second-order diffusive terms. In the frame of reference of the traveling wave, the solution of a discontinuous wave consists of a diffusive layer that grows with a rate of t 1/2 , yielding a convergence rate of 1/2. Self-similar heuristics for higher order discretizations produce a growth rate for the layer thickness of ∆t 1/(p+1) which yields an estimate for the convergence rate as p/(p+1) where p is the order of the discretization. In this paper we show that this estimated convergence rate can be derived with greater rigor for both dissipative and dispersive forms of the discrete error. In particular, the form of the analytical solution for linear modified equations can be solved exactly. These estimates and forms for the error are confirmed in a variety of demonstrations ranging from simple linear waves to multidimensional solutions of the Euler equations.