Formulation, analysis and numerical study of an optimization-based conservative interpolation (remap) of scalar fields for arbitrary Lagrangian–Eulerian methods

Monotonicity in high‐order curvilinear finite element arbitrary Lagrangian–Eulerian remap

“…This measure, associated with the DOF and denoted as μ( ρ), is such that μ i ( ρ) 2 [0, 1] should vanish (μ = 0) where the monotonicity bounds (17) are not violated, and only be active ( μ = 1) where (17) does not hold. This measure, associated with the DOF and denoted as μ( ρ), is such that μ i ( ρ) 2 [0, 1] should vanish (μ = 0) where the monotonicity bounds (17) are not violated, and only be active ( μ = 1) where (17) does not hold.…”

“…The OBR approach is very general and has been applied successfully in low-order remap settings [17] and is similar in nature to the 'repair' approach in [18]. However, the fact that it is not based on a semi-discrete formulation has the potential drawback that physical intuition might be difficult to express in terms of formal constraints.…”

“…First, we need a good 'monotonicity measure' to inform our choice as to where to apply the discrete upwinding. This measure, associated with the DOF and denoted as μ( ρ), is such that μ i ( ρ) 2 [0, 1] should vanish (μ = 0) where the monotonicity bounds (17) are not violated, and only be active ( μ = 1) where (17) does not hold. A straightforward way to define μ( ρ) is to perform a high-order, linear advection step based on (10), and then based on the high-order solution ρ, compute…”

“…Our third method for ensuring monotonicity consists of two main steps: (i) given a vector ρ ≈ ρ(τ), compute a high-order approximation ρ H ≈ ρ(τ + Δτ) to problem (10), using, for example, an explicit multistage RK method; (ii) define the new density vector ρ new ≈ ρ(τ + Δτ) by solving an optimization problem that ensures the following properties for ρ new : it is close to ρ H , has the same mass as ρ H and satisfies the monotonicity bounds (17). Specifically, in step 2, we use the massvariable mass-target (MVMT) version of the OBR algorithm; see [5,15,16].…”

“…; and ρ min , ρ max are the vectors of local lower and upper bounds in (17). This problem can be solved efficiently [16], leading to a fast one step method that is both conservative and LED monotonic.…”