diff --git a/assets/figure/cavity.png b/assets/figure/cavity.png
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diff --git a/assets/mesh/linesource.su2 b/assets/mesh/linesource.su2
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index 00000000..e2d925da
--- /dev/null
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diff --git a/example/gas/sod.jl b/example/gas/sod.jl
index 7342d5b4..d11ba8ac 100644
--- a/example/gas/sod.jl
+++ b/example/gas/sod.jl
@@ -31,3 +31,24 @@ res = zeros(3)
 end
 
 plot(ks, ctr)
+
+#--- pure 1D setting ---#
+set = Setup(case = "sod", space = "1d1f1v", maxTime = 0.12)
+gas = Gas(Kn = 1e-4, K = 0.0, γ = 3.0)
+fw, ff, bc, p = config_ib(set, ps, vs, gas)
+ib = IB1F(fw, ff, bc, p)
+
+ks = SolverSet(set, ps, vs, gas, ib)
+ctr, face = init_fvm(ks, :static_array; structarray = true)
+
+t = 0.0
+dt = KitBase.timestep(ks, ctr, t)
+nt = ks.set.maxTime ÷ dt |> Int
+res = zeros(3)
+@showprogress for iter = 1:nt
+    reconstruct!(ks, ctr)
+    evolve!(ks, ctr, face, dt)
+    update!(ks, ctr, face, dt, res)
+end
+
+plot(ks, ctr)
diff --git a/example/radiation/linesource_mn.jl b/example/radiation/linesource_mn.jl
new file mode 100644
index 00000000..f984e3b1
--- /dev/null
+++ b/example/radiation/linesource_mn.jl
@@ -0,0 +1,172 @@
+using KitBase, LinearAlgebra
+using KitML.Solaris.Optim
+using KitBase.ProgressMeter: @showprogress
+
+# one-cell simplification
+begin
+    quadratureorder = 5
+    points, weights = KitBase.octa_quadrature(quadratureorder)
+    nq = size(points, 1)
+    L = 1
+    ne = (L + 1)^2
+
+    α = zeros(ne)
+    u0 = [2.0, 0.0, 0.0, 0.0]
+    m = KitBase.eval_spherharmonic(points, L)
+
+    res = KitBase.optimize_closure(α, m, weights, u0, KitBase.maxwell_boltzmann_dual)
+    u = KitBase.realizable_reconstruct(
+        res.minimizer,
+        m,
+        weights,
+        KitBase.maxwell_boltzmann_dual_prime,
+    )
+end
+
+# multi-cell case
+begin
+    # space
+    x0 = -1.5
+    x1 = 1.5
+    y0 = -1.5
+    y1 = 1.5
+    nx = 50#100
+    ny = 50#100
+    dx = (x1 - x0) / nx
+    dy = (y1 - y0) / ny
+    pspace = KitBase.PSpace2D(x0, x1, nx, y0, y1, ny)
+
+    # time
+    tEnd = 1.0
+    cfl = 0.5
+    dt = cfl / 2 * (dx * dy) / (dx + dy)
+
+    # quadrature
+    quadratureorder = 5
+    points, weights = KitBase.octa_quadrature(quadratureorder)
+    nq = size(points, 1)
+    vspace = KitBase.VSpace1D{Float64,Int64,typeof(points),typeof(weights)}(
+        -1.0,
+        1.0,
+        nq,
+        points,
+        zero(points),
+        weights,
+    )
+
+    # particle
+    SigmaS = 1.0 .* ones(ny, nx)
+    SigmaA = 0.0 .* ones(ny, nx)
+    SigmaT = SigmaS + SigmaA
+
+    # moments
+    L = 1
+    ne = (L + 1)^2
+    phi = zeros(ne, nx, ny)
+    α = zeros(ne, nx, ny)
+    m = KitBase.eval_spherharmonic(points, L)
+end
+
+# initial distribution
+begin
+    s2 = 0.03^2
+    flr = 1e-4
+    init_field(x, y) = max(flr, 1.0 / (4.0 * pi * s2) * exp(-(x^2 + y^2) / 4.0 / s2))
+    for j = 1:nx
+        for i = 1:ny
+            y = y0 + (i - 0.5) * dy
+            x = x0 + (j - 0.5) * dx
+            # only zeroth order moment is non-zero
+            phi[1, i, j] = init_field(x, y)
+        end
+    end
+end
+
+global t = 0.0
+flux1 = zeros(ne, nx + 1, ny)
+flux2 = zeros(ne, nx, ny + 1)
+
+@showprogress for iter = 1:20
+    # regularization
+    Threads.@threads for j = 1:ny
+        for i = 1:nx
+            res = KitBase.optimize_closure(
+                α[:, i, j],
+                m,
+                vspace.weights,
+                phi[:, i, j],
+                KitBase.maxwell_boltzmann_dual,
+            )
+            α[:, i, j] .= res.minimizer
+
+            phi[:, i, j] .= KitBase.realizable_reconstruct(
+                res.minimizer,
+                m,
+                vspace.weights,
+                KitBase.maxwell_boltzmann_dual_prime,
+            )
+        end
+    end
+
+    # flux
+    fη1 = zeros(nq)
+    for j = 1:ny
+        for i = 2:nx
+            KitBase.flux_kfvs!(
+                fη1,
+                KitBase.maxwell_boltzmann_dual.(α[:, i-1, j]' * m)[:],
+                KitBase.maxwell_boltzmann_dual.(α[:, i, j]' * m)[:],
+                vspace.u[:, 1],
+                dt,
+            )
+
+            for k in axes(flux1, 1)
+                flux1[k, i, j] = sum(m[k, :] .* vspace.weights .* fη1)
+            end
+        end
+    end
+
+    fη2 = zeros(nq)
+    for i = 1:nx
+        for j = 2:ny
+            KitBase.flux_kfvs!(
+                fη2,
+                KitBase.maxwell_boltzmann_dual.(α[:, i, j-1]' * m)[:],
+                KitBase.maxwell_boltzmann_dual.(α[:, i, j]' * m)[:],
+                vspace.u[:, 2],
+                dt,
+            )
+
+            for k in axes(flux2, 1)
+                flux2[k, i, j] = sum(m[k, :] .* (vspace.weights .* fη2))
+            end
+        end
+    end
+
+    # update
+    for j = 2:ny-1
+        for i = 2:nx-1
+            for q = 1:1
+                phi[q, i, j] =
+                    phi[q, i, j] +
+                    (flux1[q, i, j] - flux1[q, i+1, j]) / dx +
+                    (flux2[q, i, j] - flux2[q, i, j+1]) / dy +
+                    (SigmaS[i, j] * phi[q, i, j] - SigmaT[i, j] * phi[q, i, j]) * dt
+            end
+
+            for q = 2:ne
+                phi[q, i, j] =
+                    phi[q, i, j] +
+                    (flux1[q, i, j] - flux1[q, i+1, j]) / dx +
+                    (flux2[q, i, j] - flux2[q, i, j+1]) / dy +
+                    (-SigmaT[i, j] * phi[q, i, j]) * dt
+            end
+        end
+    end
+
+    global t += dt
+end
+
+using Plots
+contourf(pspace.x[1:nx, 1], pspace.y[1, 1:ny], α[1, :, :])
+contourf(pspace.x[1:nx, 1], pspace.y[1, 1:ny], phi[1, :, :])
diff --git a/example/radiation/linesource.jl b/example/radiation/linesource_sn_str.jl
similarity index 95%
rename from example/radiation/linesource.jl
rename to example/radiation/linesource_sn_str.jl
index d8cc5197..480aae4e 100644
--- a/example/radiation/linesource.jl
+++ b/example/radiation/linesource_sn_str.jl
@@ -24,8 +24,8 @@ begin
     nq = size(points, 1)
 
     # particle
-    SigmaS = 1 * ones(ny + 4, nx + 4)
-    SigmaA = 0 * ones(ny + 4, nx + 4)
+    SigmaS = 1.0 .* ones(ny + 4, nx + 4)
+    SigmaA = 0.0 .* ones(ny + 4, nx + 4)
     SigmaT = SigmaS + SigmaA
 end
 
diff --git a/example/radiation/linesource_sn_unstr.jl b/example/radiation/linesource_sn_unstr.jl
new file mode 100644
index 00000000..e2a22e0b
--- /dev/null
+++ b/example/radiation/linesource_sn_unstr.jl
@@ -0,0 +1,126 @@
+using KitBase, LinearAlgebra
+using KitBase.ProgressMeter: @showprogress
+
+cd(@__DIR__)
+ps = KitBase.UnstructPSpace("../../assets/mesh/linesource.su2")
+
+begin
+    # quadrature
+    quadratureorder = 10
+    points, weights = KitBase.legendre_quadrature(quadratureorder)
+    nq = size(points, 1)
+    vs = KitBase.UnstructVSpace(-1.0, 1.0, nq, points, weights)
+
+    # IC
+    s2 = 0.03^2
+    init_field(x, y) = max(1e-4, 1.0 / (4.0 * π * s2) * exp(-(x^2 + y^2) / 4.0 / s2))
+
+    # particle
+    SigmaS = ones(size(ps.cellid, 1))
+    SigmaA = zeros(size(ps.cellid, 1))
+    SigmaT = SigmaS + SigmaA
+
+    # time
+    tspan = (0.0, 0.3)
+    cfl = 0.7
+end
+
+ctr = Array{KitBase.ControlVolumeUS1F}(undef, size(ps.cellid, 1))
+for i in eachindex(ctr)
+    n = Vector{Float64}[]
+    for j = 1:3
+        push!(
+            n,
+            KitBase.unit_normal(
+                ps.points[ps.facePoints[ps.cellFaces[i, j], 1], :],
+                ps.points[ps.facePoints[ps.cellFaces[i, j], 2], :],
+            ),
+        )
+
+        if dot(ps.faceCenter[ps.cellFaces[i, j], :] .- ps.cellCenter[i, :], n[j]) < 0
+            n[j] .= -n[j]
+        end
+    end
+
+    phi = zeros(nq)
+    phi .= init_field(ps.cellCenter[i, 1], ps.cellCenter[i, 2])
+
+    w = sum(weights .* phi)
+    dx = [
+        KitBase.point_distance(
+            ps.cellCenter[i, :],
+            ps.points[ps.cellid[i, 1], :],
+            ps.points[ps.cellid[i, 2], :],
+        ),
+        KitBase.point_distance(
+            ps.cellCenter[i, :],
+            ps.points[ps.cellid[i, 2], :],
+            ps.points[ps.cellid[i, 3], :],
+        ),
+        KitBase.point_distance(
+            ps.cellCenter[i, :],
+            ps.points[ps.cellid[i, 3], :],
+            ps.points[ps.cellid[i, 1], :],
+        ),
+    ]
+
+    ctr[i] = KitBase.ControlVolumeUS1F(n, ps.cellCenter[i, :], dx, w, w, phi)
+end
+
+face = Array{KitBase.Interface2D1F}(undef, size(ps.facePoints, 1))
+for i in eachindex(face)
+    len = norm(ps.points[ps.facePoints[i, 1], :] .- ps.points[ps.facePoints[i, 2], :])
+    n = KitBase.unit_normal(ps.points[ps.facePoints[i, 1], :], ps.points[ps.facePoints[i, 2], :])
+
+    if !(-1 in ps.faceCells[i, :])
+        n0 = ps.cellCenter[ps.faceCells[i, 2], :] .- ps.cellCenter[ps.faceCells[i, 1], :]
+    else
+        n0 = zero(n)
+    end
+    if dot(n, n0) < 0
+        n .= -n
+    end
+
+    fw = 0.0
+    ff = zeros(nq)
+
+    face[i] = KitBase.Interface2D1F(len, n[1], n[2], fw, ff)
+end
+
+dt = 1.2 / 150 * cfl
+nt = tspan[2] ÷ dt |> Int
+@showprogress for iter = 1:nt
+    @inbounds Threads.@threads for i in eachindex(face)
+        velo = vs.u[:, 1] .* face[i].n[1] + vs.u[:, 2] .* face[i].n[2]
+        if !(-1 in ps.faceCells[i, :])
+            KitBase.flux_kfvs!(
+                face[i].ff,
+                ctr[ps.faceCells[i, 1]].f,
+                ctr[ps.faceCells[i, 2]].f,
+                velo,
+                dt,
+            )
+        end
+    end
+
+    @inbounds Threads.@threads for i in eachindex(ctr)
+        if ps.cellType[i] == 0
+            for j = 1:3
+                dirc = sign(dot(ctr[i].n[j], face[ps.cellFaces[i, j]].n))
+                @. ctr[i].f -=
+                    dirc * face[ps.cellFaces[i, j]].ff * face[ps.cellFaces[i, j]].len /
+                    ps.cellArea[i]
+            end
+
+            integral = KitBase.discrete_moments(ctr[i].f, vs.weights)
+            integral /= 4.0 * π
+            @. ctr[i].f += (integral - ctr[i].f) * dt
+        end
+    end
+end
+
+cdata = zeros(length(ctr), 1)
+for i in eachindex(ctr)
+    cdata[i, 1] = sum(ctr[i].f .* vs.weights)
+end
+write_vtk(ps.points, ps.cellid, cdata)