Some schemes, such as those with upwind FDs, are intrinsically dissipative, with a fixed “amount” of dissipation for a given resolution. Another approach is to add to the discretization a dissipative operator with a tunable strength factor ,
The main property that sometimes allows numerical dissipation to stabilize otherwise unstable schemes is when they strictly carry away energy (as in the energy definitions involved in well-posedness or numerical-stability analysis) from the system. For example, the operators (8.53) are semi-negative definite
In the presence of boundaries, it is standard to simply set the operators (8.53) to zero near them. The result is, in general, not semi-negative definite as in (8.54), which cannot only not help resolve instabilities but also trigger them. Many times this is not the case in practice if the boundary is an outer one, where the solution is weak, but not for inter-domain boundaries (see Section 10). For example, for a discretization of the standard wave equation on a multi-domain, curvilinear grid setting, using the SBP operator with Kreiss–Oliger dissipation set to zero near interpatch boundaries does not lead to stability while the more elaborate construction below does .
For SBP-based schemes, adding artificial dissipation may lead to an unstable scheme unless the dissipation operator is semi-negative under the SBP scalar product. In addition, the dissipation operator should ideally be non-vanishing all the way up to the boundary and preserve the accuracy of the scheme everywhere (which is more difficult in the SBP case, as it is non-uniform). In , a prescription for operators satisfying both conditions for arbitrary–high-order SBP scalar products, is presented. A compatible dissipation operator is constructed asboundary operator. The latter has to be positive semi-definite and its role is to allow boundary points to be treated differently from interior points. cannot be chosen freely, but has to follow certain restrictions (which become somewhat involved in the non-diagonal SBP case) based on preserving the accuracy of the schemes near and at boundaries; see  for more details.
Living Rev. Relativity 15, (2012), 9
This work is licensed under a Creative Commons License.