Small performance optimizations

[efaulhaber]

This PR contains two small performance optimizations. The first is an algebraic simplification of the derivatives of the Wendland kernels and the normalization factors:

main:

julia> r = rand(SVector{3, Float64}); d = norm(r); h = 1.0; kernel = WendlandC2Kernel{3}();

julia> @b TrixiParticles.kernel_grad($kernel, $r, $d, $h) seconds=1
3.409 ns

julia> r = rand(SVector{2, Float64}); d = norm(r); h = 1.0; kernel = WendlandC2Kernel{2}();

julia> @b TrixiParticles.kernel_grad($kernel, $r, $d, $h) seconds=1
2.650 ns

With result simplified algebraically:

julia> r = rand(SVector{3, Float64}); d = norm(r); h = 1.0; kernel = WendlandC2Kernel{3}();

julia> @b TrixiParticles.kernel_grad($kernel, $r, $d, $h) seconds=1
3.006 ns

julia> r = rand(SVector{2, Float64}); d = norm(r); h = 1.0; kernel = WendlandC2Kernel{2}();

julia> @b TrixiParticles.kernel_grad($kernel, $r, $d, $h) seconds=1
2.573 ns

With normalization_factor simplified to 7 / (pi * h^2 * 4):

julia> r = rand(SVector{3, Float64}); d = norm(r); h = 1.0; kernel = WendlandC2Kernel{3}();

julia> @b TrixiParticles.kernel_grad($kernel, $r, $d, $h) seconds=1
2.843 ns

julia> r = rand(SVector{2, Float64}); d = norm(r); h = 1.0; kernel = WendlandC2Kernel{2}();

julia> @b TrixiParticles.kernel_grad($kernel, $r, $d, $h) seconds=1
2.557 ns

Interestingly, this difference is not measurable when benchmarking only kernel_deriv.

The second is a small optimization of the computation of v_max (apparently only relevant on the CPU).

julia> A = rand(3, 10_000_000);

julia> @b maximum(x -> sqrt(dot(x, x)), reinterpret(reshape, SVector{3, eltype($A)}, view($A, 1:3, :))) seconds=1
6.966 ms

julia> @b sqrt(maximum(x -> dot(x, x), reinterpret(reshape, SVector{3, eltype($A)}, view($A, 1:3, :)))) seconds=1
6.574 ms

julia> A = Metal.rand(3, 10_000_000);

julia> @b maximum(x -> sqrt(dot(x, x)), reinterpret(reshape, SVector{3, eltype($A)}, view($A, 1:3, :))) seconds=1
982.000 μs (664 allocs: 15.070 KiB)

julia> @b sqrt(maximum(x -> dot(x, x), reinterpret(reshape, SVector{3, eltype($A)}, view($A, 1:3, :)))) seconds=1
981.459 μs (664 allocs: 15.070 KiB)

[Copilot]

Pull Request Overview

This PR refactors smoothing kernel normalization factors and related computations to improve GPU performance and code clarity. The changes focus on simplifying arithmetic expressions to reduce instructions and improve readability.

Key Changes:

• Simplified normalization factor expressions by consolidating divisions (e.g., a / b / c → a / (b * c))
• Optimized v_max computation in particle shifting to compute squared magnitude first, then take square root
• Simplified kernel derivative formulas for Wendland kernels by algebraically reducing expressions

Reviewed Changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 2 comments.

File	Description
src/schemes/fluid/shifting_techniques.jl	Optimized v_max calculation to compute maximum of squared velocities before taking square root
src/general/smoothing_kernels.jl	Simplified normalization factors and kernel derivatives across multiple kernel types (Schoenberg, Wendland, Poly6)

---

💡 Add Copilot custom instructions for smarter, more guided reviews. Learn how to get started.

[efaulhaber]

/run-gpu-tests