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 [141
].
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 [303], a prescription for
operators satisfying both conditions for arbitrary–high-order SBP scalar products, is presented. A
compatible dissipation operator is constructed as
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Living Rev. Relativity 15, (2012), 9
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