Numerical experiments show, for the velocity variable, second-order convergence in a discrete L2 norm and first-order convergence in a discrete H1 norm.
Since the pseudospectral differentiation is taken at a point-wise level, a discrete (L^{2}) norm and inner product need to be introduced to facilitate the analysis.
where the constant normalizes the discrete norm of to 1.
For any discrete function v i, we also define the corresponding discrete norm by ∥ v ∥ ∞, ω h ≡ ∥ v ∥ ∞ = max 1 ≤ i ≤ N | v i |.
The discrete norm (|cdot|_{2}) of numerical errors at (T=5) is given in Figure 2, which shows apparently the spatial spectral accuracy is verified.
The new norm in the space (V_{h}) is defined as Vert v_{h} Vert ^{2}_{h}=sum _{Tinmathcal{T}_{h}} vert v_{h} vert ^{2}_{1,T} +sum_{Einmathcal{E}_{h}} biggl{ frac{1}{h_{E}} biglVert [[v_{h}]] bigrVert ^{2}_{0,E} biggr}, (2.6) which is larger than the traditional discrete norm.
We establish global error estimates to verify the second-order temporal accuracy of the proposed method for velocity, pressure, and temperature in terms of a discrete l2-norm.
Thus a greater concentration on the discrete norms of contract planning and 'completeness' – by attempting to draft a 'hard' contract that builds in appropriate incentives, fixes responsibilities, specifies what happens when unexpected events arise, and is rigorously enforced – might be predicted to lead to conflictual or adversarial relations with high costs to both parties.
This reformulation leads to proofs of conservation and stability in discrete norms associated with the method, recovering the linearly stable CPR schemes of Vincent et al. (2011, 2015).
According to the definitions of the discrete norms, the energy can be represented as (E_{h}(t)=frac{1}{2}(|vec{phi}'_{h}(t)|_{0}+|vec {phi}_{h}(t)|_{1})), where (vec{phi}_{h}(t)) is the solution of equation (1.4).
