On the convergence rates of energy-stable finite-difference schemes

Abstract We consider constant-coefficient initial-boundary value problems, with a first or second derivative in time and a highest spatial derivative of order q, and their semi-discrete finite difference approximations. With an internal truncation error of order p ≥ 1 , and a boundary error of order r ≥ 0 , we prove that the convergence rate is: min ⁡ ( p , r + q ) . The assumptions needed for these results to hold are: i) The continuous problem is linear and well-posed (with a smooth solution). ii) The numerical scheme is consistent, nullspace consistent, nullspace invariant, and energy stable. These assumptions are often satisfied for Summation-By-Parts schemes.

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