From eeaf318be2041eb93183bba325c103df2476f0d8 Mon Sep 17 00:00:00 2001
From: Florian Atteneder <florian.atteneder@uni-jena.de>
Date: Fri, 6 Sep 2024 20:45:22 +0000
Subject: [PATCH] dg1d: Add limiters.jl
 (https://git.tpi.uni-jena.de/dg/dg1d.jl/-/merge_requests/221)

- Collect most of the limiter implementations into `limiters.jl` (there still remain bits in `ScalarEq`)
- Add new limiter `limit_slope_preserve_continuity!` that is tailored towards the Bspline approximation (although its not working well)
- Add new limiter `limit_slope_convex_hull!` that is tailored towards the Bspline approximation
- Add new reftests `burgers_sine_bspline1_limit_slope_convex_hull` and ``burgers_sine_bspline2_limit_slope_convex_hull`
- Fix Bspline integration
---
 src/EulerEq/rhs.jl                            |  24 +-
 src/GRHD/rhs.jl                               |  28 +--
 src/HRSC/mda.jl                               |   2 +-
 src/SRHD/rhs.jl                               |  24 +-
 src/ScalarEq/callbacks.jl                     |   3 +-
 src/ScalarEq/rhs.jl                           |  39 +--
 src/ScalarEq/setup.jl                         |   3 +-
 src/TCI/minmod.jl                             |   4 +-
 src/TCI/modaldecayaverage.jl                  |   2 +-
 src/bspline.jl                                |   8 +-
 src/dg1d.jl                                   |   2 +
 src/limiters.jl                               | 222 ++++++++++++++++++
 src/mesh.jl                                   |  56 -----
 ...sine_bspline1_limit_slope_convex_hull.toml |  23 ++
 .../output1d.h5                               | Bin 0 -> 33032 bytes
 ...sine_bspline2_limit_slope_convex_hull.toml |  23 ++
 .../output1d.h5                               | Bin 0 -> 33928 bytes
 test/IntegrationTests/refs/testconfig.toml    |   5 +
 18 files changed, 302 insertions(+), 166 deletions(-)
 create mode 100644 src/limiters.jl
 create mode 100644 test/IntegrationTests/refs/burgers_sine_bspline1_limit_slope_convex_hull/burgers_sine_bspline1_limit_slope_convex_hull.toml
 create mode 100644 test/IntegrationTests/refs/burgers_sine_bspline1_limit_slope_convex_hull/output1d.h5
 create mode 100644 test/IntegrationTests/refs/burgers_sine_bspline2_limit_slope_convex_hull/burgers_sine_bspline2_limit_slope_convex_hull.toml
 create mode 100644 test/IntegrationTests/refs/burgers_sine_bspline2_limit_slope_convex_hull/output1d.h5

diff --git a/src/EulerEq/rhs.jl b/src/EulerEq/rhs.jl
index 2d788945..f8891df8 100644
--- a/src/EulerEq/rhs.jl
+++ b/src/EulerEq/rhs.jl
@@ -203,7 +203,7 @@ function step_fv_muscl!(P::Project, mesh::Mesh1d{FVElement}, state)
 
     # limit slope and compute left (L) and (R) interface values
     for k in 1:K
-      dvL = limit_slope(dvâ‚Š[k], dvâ‚‹[k], dl, prms_slope_limiter)
+      dvL = dg1d.limit_slope(dvâ‚Š[k], dvâ‚‹[k], dl, prms_slope_limiter)
       vL[k] = v[k] - dvL/2
       vR[k] = v[k] + dvL/2
     end
@@ -317,28 +317,6 @@ function step_fv_muscl!(P::Project, mesh::Mesh1d{FVElement}, state)
 end
 
 
-@inline function limit_slope(a, b, dx, prms)
-  if prms.method === :none
-    return 0.0
-  elseif prms.method === :minmod
-    return TCI.minmod(a, b)
-  elseif prms.method === :muscl
-    return TCI.minmod((a+b)/2, 2*a, 2*b)
-  elseif prms.method === :superbee
-    return TCI.minmod(TCI.maxmod(a,b),TCI.minmod(2*a,2*b))
-  elseif prms.method === :van_albada
-    c = dx^3
-    return TCI.minmod( ((a^2+c^2)*b+(b^2+c^2)*a)/(a^2+b^2+c^2), 2*a, 2*b )
-  elseif prms.method === :van_leer
-    return TCI.minmod(2*a*b/(a+b), 2*a, 2*b)
-  elseif prms.method === :tvb
-    return TCI.minmod_M(dx, prms.tvb_M, (a+b)/2, 2*a, 2*b)
-  else
-    TODO(prms.method)
-  end
-end
-
-
 #######################################################################
 #                               DG rhs                                #
 #######################################################################
diff --git a/src/GRHD/rhs.jl b/src/GRHD/rhs.jl
index 09187952..6a1b8636 100644
--- a/src/GRHD/rhs.jl
+++ b/src/GRHD/rhs.jl
@@ -452,7 +452,7 @@ function step_fv_slope_limiter!(mesh, P::Project{:spherical1d}, state)
       kâ‚‹, kâ‚Š = k-1, k+1
       wâ‚‹, wâ‚€, wâ‚Š = w[kâ‚‹], w[k], w[kâ‚Š]
       dwâ‚Š[k], dwâ‚‹[k] = wâ‚Š-wâ‚€, wâ‚€-wâ‚‹
-      dwL = limit_slope(dwâ‚Š[k], dwâ‚‹[k], dl, prms_slope_limiter)
+      dwL = dg1d.limit_slope(dwâ‚Š[k], dwâ‚‹[k], dl, prms_slope_limiter)
       wL[k] = w[k] - dwL/2
       wR[k] = w[k] + dwL/2
     end
@@ -725,7 +725,7 @@ function step_fv_slope_limiter!(mesh, P::Project{:doublecartoon}, state)
       kâ‚‹, kâ‚Š = k-1, k+1
       wâ‚‹, wâ‚€, wâ‚Š = w[kâ‚‹], w[k], w[kâ‚Š]
       dwâ‚Š[k], dwâ‚‹[k] = wâ‚Š-wâ‚€, wâ‚€-wâ‚‹
-      dwL = limit_slope(dwâ‚Š[k], dwâ‚‹[k], dl, prms_slope_limiter)
+      dwL = dg1d.limit_slope(dwâ‚Š[k], dwâ‚‹[k], dl, prms_slope_limiter)
       wL[k] = w[k] - dwL/2
       wR[k] = w[k] + dwL/2
     end
@@ -1129,30 +1129,6 @@ end
 end
 
 
-@inline function limit_slope(a, b, dx, prms)
-  if prms.method === :none
-    return 0.0
-  elseif prms.method === :minmod
-    return TCI.minmod(a, b)
-  elseif prms.method === :muscl
-    return TCI.minmod((a+b)/2, 2*a, 2*b)
-  elseif prms.method === :superbee
-    return TCI.minmod(TCI.maxmod(a,b),TCI.minmod(2*a,2*b))
-  elseif prms.method === :van_albada
-    c = dx^3
-    return TCI.minmod( ((a^2+c^2)*b+(b^2+c^2)*a)/(a^2+b^2+c^2), 2*a, 2*b )
-  elseif prms.method === :van_leer
-    return TCI.minmod(2*a*b/(a+b), 2*a, 2*b)
-  elseif prms.method === :tvb
-    return TCI.minmod_M(dx, prms.tvb_M, (a+b)/2, 2*a, 2*b)
-  elseif prms.method === :mc
-    return dg1d.MC(b, a)
-  else
-    TODO(prms.method)
-  end
-end
-
-
 #######################################################################
 #                               DG RHS                                #
 #######################################################################
diff --git a/src/HRSC/mda.jl b/src/HRSC/mda.jl
index a85e51f2..18779814 100644
--- a/src/HRSC/mda.jl
+++ b/src/HRSC/mda.jl
@@ -88,7 +88,7 @@ function compute_viscosity!(
     @. data_y = log(skylined_mod_uh)
     @. data_y = max(data_y, -20.0)
     _, _, tau = dg1d.least_sqr_linear(data_x, data_y)
-    isnan(tau) && error("undesired nan encountered")
+    !isfinite(tau) && error("undesired nan encountered")
 
     mu[j] *= v[j]
     if tau > n_high
diff --git a/src/SRHD/rhs.jl b/src/SRHD/rhs.jl
index 271cf2ff..58ac2239 100644
--- a/src/SRHD/rhs.jl
+++ b/src/SRHD/rhs.jl
@@ -221,7 +221,7 @@ function step_fv_muscl!(P::Project, mesh::Mesh1d{FVElement}, state)
 
     # limit slope and compute left (L) and (R) interface values
     for k in 1:K
-      dwL = limit_slope(dwâ‚Š[k], dwâ‚‹[k], dl, prms_slope_limiter)
+      dwL = dg1d.limit_slope(dwâ‚Š[k], dwâ‚‹[k], dl, prms_slope_limiter)
       wL[k] = w[k] - dwL/2
       wR[k] = w[k] + dwL/2
     end
@@ -334,28 +334,6 @@ function step_fv_muscl!(P::Project, mesh::Mesh1d{FVElement}, state)
 end
 
 
-@inline function limit_slope(a, b, dx, prms)
-  if prms.method === :none
-    return 0.0
-  elseif prms.method === :minmod
-    return TCI.minmod(a, b)
-  elseif prms.method === :muscl
-    return TCI.minmod((a+b)/2, 2*a, 2*b)
-  elseif prms.method === :superbee
-    return TCI.minmod(TCI.maxmod(a,b),TCI.minmod(2*a,2*b))
-  elseif prms.method === :van_albada
-    c = dx^3
-    return TCI.minmod( ((a^2+c^2)*b+(b^2+c^2)*a)/(a^2+b^2+c^2), 2*a, 2*b )
-  elseif prms.method === :van_leer
-    return TCI.minmod(2*a*b/(a+b), 2*a, 2*b)
-  elseif prms.method === :tvb
-    return TCI.minmod_M(dx, prms.tvb_M, (a+b)/2, 2*a, 2*b)
-  else
-    TODO(prms.method)
-  end
-end
-
-
 #######################################################################
 #                               DG rhs                                #
 #######################################################################
diff --git a/src/ScalarEq/callbacks.jl b/src/ScalarEq/callbacks.jl
index 49af0837..48a1f43d 100644
--- a/src/ScalarEq/callbacks.jl
+++ b/src/ScalarEq/callbacks.jl
@@ -246,7 +246,6 @@ end
 #######################################################################
 
 
-# TODO
 callback_analyze(state_u, state_t, env, P, mesh::Mesh) = nothing
 
 
@@ -255,6 +254,8 @@ function callback_analyze(state_u, state_t, env, P, mesh::Mesh1d)
   @unpack u_norm2 = get_global_variables(mesh)
   u_norm2 .= dg1d.norm_L2(u, mesh)^2
 end
+# TODO Make this use dg1d.limit_slope!
+# after moving src/ScalarEq/rhs.jl:limit_slope! to src/limters.jl
 function callback_analyze(state_u, state_t, env, P, mesh::Mesh1d{<:FVElement})
   @unpack u = get_dynamic_variables(mesh)
   @unpack u_norm2 = get_global_variables(mesh)
diff --git a/src/ScalarEq/rhs.jl b/src/ScalarEq/rhs.jl
index 9cd9b0e5..e1f3cc61 100644
--- a/src/ScalarEq/rhs.jl
+++ b/src/ScalarEq/rhs.jl
@@ -132,6 +132,7 @@ function rhs_fv_muscl!(env, mesh::Mesh1d{FVElement}, P::Project)
 end
 
 
+# TODO Move this to src/limiters.jl
 function limit_slope!(P::Project, mesh::Mesh1d{FVElement}, u)
   @unpack equation = P
   @unpack uL, uR,
@@ -158,7 +159,7 @@ function limit_slope!(P::Project, mesh::Mesh1d{FVElement}, u)
 
   # limit slope and compute left (L) and (R) interface values
   for k in 1:K
-    duL = limit_slope(duâ‚Š[k], duâ‚‹[k], dl, prms_slope_limiter)
+    duL = dg1d.limit_slope(duâ‚Š[k], duâ‚‹[k], dl, prms_slope_limiter)
     uL[k] = u[k] - duL/2
     uR[k] = u[k] + duL/2
   end
@@ -240,28 +241,6 @@ function step_fv_muscl!(P::Project, mesh::Mesh1d{FVElement}, u)
 end
 
 
-@inline function limit_slope(a, b, dx, prms)
-  if prms.method === :none
-    return 0.0
-  elseif prms.method === :minmod
-    return TCI.minmod(a, b)
-  elseif prms.method === :muscl
-    return TCI.minmod((a+b)/2, 2*a, 2*b)
-  elseif prms.method === :superbee
-    return TCI.minmod(TCI.maxmod(a,b),TCI.minmod(2*a,2*b))
-  elseif prms.method === :van_albada
-    c = dx^3
-    return TCI.minmod( ((a^2+c^2)*b+(b^2+c^2)*a)/(a^2+b^2+c^2), 2*a, 2*b )
-  elseif prms.method === :van_leer
-    return TCI.minmod(2*a*b/(a+b), 2*a, 2*b)
-  elseif prms.method === :tvb
-    return TCI.minmod_M(dx, prms.tvb_M, (a+b)/2, 2*a, 2*b)
-  else
-    TODO(prms.method)
-  end
-end
-
-
 #######################################################################
 #                               DG RHS                                #
 #######################################################################
@@ -269,14 +248,14 @@ end
 
 function rhs!(env, mesh::Mesh1d, P::Project, hrsc::Nothing)
   if mesh.element.kind ∈ (:modal_bspline2, :modal_bspline1)
-    modal_rhs!(env, mesh, P, hrsc)
+    modal_rhs!(env, mesh, P)
   else
-    nodal_rhs!(env, mesh, P, hrsc)
+    nodal_rhs!(env, mesh, P)
   end
 end
 
 
-function nodal_rhs!(env, mesh::Mesh1d, P::Project, hrsc::Nothing)
+function nodal_rhs!(env, mesh::Mesh1d, P::Project)
   @unpack flx_u, src_u   = get_static_variables(mesh)
   @unpack u              = get_dynamic_variables(mesh)
   @unpack rhs_u          = get_rhs_variables(mesh)
@@ -291,7 +270,7 @@ function nodal_rhs!(env, mesh::Mesh1d, P::Project, hrsc::Nothing)
 end
 
 
-function modal_rhs!(env, mesh::Mesh1d, P::Project, hrsc::Nothing)
+function modal_rhs!(env, mesh::Mesh1d, P::Project)
   @unpack flx_u, src_u,
           u_modal, flx_u_modal,
           src_u_modal    = get_static_variables(mesh)
@@ -299,6 +278,11 @@ function modal_rhs!(env, mesh::Mesh1d, P::Project, hrsc::Nothing)
   @unpack rhs_u          = get_rhs_variables(mesh)
   @unpack nflx_u, bdry_u = get_bdry_variables(mesh)
 
+  if P.prms.hrsc === :slope_limiter
+    @unpack flag = get_cell_variables(mesh)
+    dg1d.limit_slope_convex_hull!(u, flag, P.prms.workspace, mesh)
+  end
+
   u_modal .= u
   for (v_u, v_u_modal) in zip(eachcell(mesh,u),eachcell(mesh,u_modal))
     mul!(v_u, P.prms.V_bspline, v_u_modal)
@@ -315,6 +299,7 @@ function modal_rhs!(env, mesh::Mesh1d, P::Project, hrsc::Nothing)
     mul!(v_src_u_modal, P.prms.invV_bspline, v_src_u)
   end
   compute_rhs_weak_form!(rhs_u, flx_u_modal, src_u_modal, nflx_u, mesh)
+
   u .= u_modal
 
   return
diff --git a/src/ScalarEq/setup.jl b/src/ScalarEq/setup.jl
index 727b4242..0d1f7926 100644
--- a/src/ScalarEq/setup.jl
+++ b/src/ScalarEq/setup.jl
@@ -24,10 +24,11 @@ function Project(env::Environment, mesh::Mesh1d, prms)
   else
     V_bspline, invV_bspline = nothing, nothing
   end
+  workspace = dg1d.Workspace()
   fixedprms = (; bernstein, av_derivative_scheme, av_drag, hrsc, muscl_omega,
                  slope_limiter_method, slope_limiter_tvb_M,
                  av_recompute_substeps,
-                 V_bernstein, V_bspline, invV_bspline,
+                 V_bernstein, V_bspline, invV_bspline, workspace,
                  analyze_error, analyze_error_norm)
 
   # construct Project
diff --git a/src/TCI/minmod.jl b/src/TCI/minmod.jl
index c87c0a01..2efdea49 100644
--- a/src/TCI/minmod.jl
+++ b/src/TCI/minmod.jl
@@ -11,8 +11,6 @@ function compute_indicator!(
     tci::Minmod)
 
   @unpack mesh       = tci
-  @unpack invdetJ    = get_static_variables(mesh.cache)
-
   Npts, K = layout(mesh)
   L,      = dg1d.widths(mesh)
   dL      = L/K
@@ -23,7 +21,7 @@ function compute_indicator!(
 
   @unpack threshold = tci
   # fill flag by applying minmod criterion
-  for k in 1:K # loop over cells
+  for k in 1:K
     # assuming periodic grid
     k_lhs, k_rhs = periodic_index(k-1, K), periodic_index(k+1, K)
     uavg, uavg_lhs, uavg_rhs = u_averages[k], u_averages[k_lhs], u_averages[k_rhs]
diff --git a/src/TCI/modaldecayaverage.jl b/src/TCI/modaldecayaverage.jl
index 12b20c65..e8bacc4e 100644
--- a/src/TCI/modaldecayaverage.jl
+++ b/src/TCI/modaldecayaverage.jl
@@ -81,7 +81,7 @@ function compute_indicator!(
     @views for m in 1:N
       uh_skylined = maximum(uh_mod[min(m,N-1):end])
       log_uh = log(uh_skylined)
-      lsq_y[m] = isinf(log_uh) ? - 20.0 : log_uh
+      lsq_y[m] = !isfinite(log_uh) ? - 20.0 : log_uh
     end
 
     # estimate effective decay rate by least square fit of modal coefficients, (3.9)
diff --git a/src/bspline.jl b/src/bspline.jl
index 3919fd54..fe51f567 100644
--- a/src/bspline.jl
+++ b/src/bspline.jl
@@ -352,7 +352,7 @@ end
 function integrate(f::AbstractVector, bs::Bspline1)
   @toggled_assert length(f) == bs.Npts
   sum = 0.0
-  @inbounds for i in bs.Npts
+  @inbounds for i in 1:bs.Npts
     fi, xl, xr = f[i], bs.xs[i], bs.xs[i+2]
     sum += fi * (xr-xl)/2
   end
@@ -361,7 +361,7 @@ end
 function integrate(f::Function, idxs::AbstractVector{<:Integer}, bs::Bspline1)
   @toggled_assert length(idxs) == bs.Npts
   sum = 0.0
-  @inbounds for i in bs.Npts
+  @inbounds for i in 1:bs.Npts
     fi, xl, xr = f(idxs[i]), bs.xs[i], bs.xs[i+2]
     sum += fi * (xr-xl)/2
   end
@@ -374,7 +374,7 @@ end
 function integrate(f::AbstractVector, bs::Bspline2)
   @toggled_assert length(f) == bs.Npts+1
   sum = 0.0
-  @inbounds for i in bs.Npts+1
+  @inbounds for i in 1:bs.Npts+1
     fi, xl, xr = f[i], bs.xs[i], bs.xs[i+3]
     sum += fi * (xr-xl)/3
   end
@@ -383,7 +383,7 @@ end
 function integrate(f::Function, idxs::AbstractVector{<:Integer}, bs::Bspline2)
   @toggled_assert length(idxs) == bs.Npts+1
   sum = 0.0
-  @inbounds for i in bs.Npts+1
+  @inbounds for i in 1:bs.Npts+1
     fi, xl, xr = f(idxs[i]), bs.xs[i], bs.xs[i+3]
     sum += fi * (xr-xl)/3
   end
diff --git a/src/dg1d.jl b/src/dg1d.jl
index 48be5deb..d52d1f48 100644
--- a/src/dg1d.jl
+++ b/src/dg1d.jl
@@ -88,6 +88,8 @@ export Mesh, Mesh1d, Mesh2d, MeshInterpolator, layout, n_points, n_cells, dimens
        min_grid_spacing
 include("mesh.jl")
 
+include("limiters.jl")
+
 export AbstractEquation
 include("equations.jl")
 
diff --git a/src/limiters.jl b/src/limiters.jl
new file mode 100644
index 00000000..c6c388ec
--- /dev/null
+++ b/src/limiters.jl
@@ -0,0 +1,222 @@
+@inline function limit_slope(a, b, dx, prms)
+  if prms.method === :none
+    return 0.0
+  elseif prms.method === :minmod
+    return TCI.minmod(a, b)
+  elseif prms.method === :muscl
+    return TCI.minmod((a+b)/2, 2*a, 2*b)
+  elseif prms.method === :superbee
+    return TCI.minmod(TCI.maxmod(a,b),TCI.minmod(2*a,2*b))
+  elseif prms.method === :van_albada
+    c = dx^3
+    return TCI.minmod( ((a^2+c^2)*b+(b^2+c^2)*a)/(a^2+b^2+c^2), 2*a, 2*b )
+  elseif prms.method === :van_leer
+    return TCI.minmod(2*a*b/(a+b), 2*a, 2*b)
+  elseif prms.method === :tvb
+    return TCI.minmod_M(dx, prms.tvb_M, (a+b)/2, 2*a, 2*b)
+  elseif prms.method === :mc
+    return dg1d.MC(b, a)
+  else
+    TODO(prms.method)
+  end
+end
+
+
+#######################################################################
+#                  slope limiters for FVElement mesh                  #
+#######################################################################
+
+
+# TODO Merge this with ScalarEq/rhs.jl:limit_slope!
+function limit_slope!(δu, u, mesh::Mesh1d{<:FVElement}, @nospecialize(prms::NamedTuple))
+  @unpack hrsc = prms
+  if hrsc === :muscl
+    limit_slope_muscl!(δu, u, mesh, prms)
+  elseif hrsc === :fv_central
+    # TODO Not sure if correct. The discussion in Hesthaven book 2018, sec 10.3 does not
+    # specify how to compute the slope. The only requirement seems to be that
+    # the slope obeys δu_j/h = du/dx(x_j) + O(h). And in the codes they provide they
+    # seem to reuse the same slope limiters as for the MUSCL scheme etc.
+    prms = (hrsc = :muscl, muscl_omega=0)
+    limit_slope_muscl!(δu, u, mesh, prms)
+  else
+    TODO(hrsc)
+  end
+end
+
+
+function limit_slope_muscl!(δu, u, mesh::Mesh1d{<:FVElement}, prms::NamedTuple)
+  # assume that FV solution in jth cell [x_(j-1/2),x_(j+1/2)] can be approximated by
+  #   u_j ≈ ̄u_j + δu_j (x-x_j)/h
+  # where δu_j is the slope at x_j and
+  #   x_j = x_j-1/2 + h/2
+  #   h   = x_j+1/2 - x_j-1/2
+  # Compare discussion in Hesthaven book 2018, sec 10.2
+  @unpack muscl_omega = prms
+  ω = muscl_omega
+  # TODO Enabling upwinding in fv_muscl_burgers_sine (ω = 1.0) gives asymmetric results.
+  # Do we need to compute two slopes like in the slope limiter implementations in EulerEq etc?
+  K, = mesh.tree.dims
+  @turbo for k = 2:K-1
+    kâ‚‹, kâ‚Š = k-1, k+1
+    uâ‚‹, uâ‚€, uâ‚Š = u[kâ‚‹], u[k], u[kâ‚Š]
+    Δ⁻u = u₀-u₋
+    Δ⁺u = u₊-u₀
+    # (10.12) in Hesthaven book 2018, sec 10.2
+    δu[k] = (1+ω)/2*Δ⁻u + (1-ω)/2*Δ⁺u
+  end
+  if isperiodic(mesh)
+    for k in (1,K)
+      kâ‚‹, kâ‚Š = periodic_index(k-1,K), periodic_index(k+1,K)
+      uâ‚‹, uâ‚€, uâ‚Š = u[kâ‚‹], u[k], u[kâ‚Š]
+      Δ⁻u = u₀-u₋
+      Δ⁺u = u₊-u₀
+      δu[k] = (1+ω)/2*Δ⁻u + (1-ω)/2*Δ⁺u
+    end
+  else
+    TODO("bdry conditions")
+  end
+end
+
+
+#######################################################################
+#            slope limiter that preserves C^0 constraints             #
+#######################################################################
+
+
+function limit_slope_preserve_continuity!(var::AbstractArray, mask::AbstractArray, ws::Workspace, mesh::Mesh1d{<:DGElement})
+  @toggled_assert mesh.element.kind === :modal_bspline1
+  enter(ws) do
+
+    @unpack x = get_static_variables(mesh)
+    Npts = mesh.element.Npts
+    K = mesh.tree.dims[1]
+    tmp_var  = borrow(ws, Npts*K)
+    avgs     = borrow(ws, Npts+1)
+    δs       = borrow(ws, Npts+1)
+    lim_avgs = borrow(ws, Npts-1)
+    lim_δs   = borrow(ws, Npts-1)
+    mat_A    = borrow(ws, (Npts-1)*(Npts-1))
+    b        = borrow(ws, Npts-1)
+    A        = reshape(mat_A,(Npts-1,Npts-1))
+
+    for k in 1:K
+      if k == 1 || k == K || mask[k] ≈ 0.0
+        # skip bdry cells, but backup data because we restore the whole vector at once later
+        idxs = cellindices(mesh, k)
+        view(tmp_var, idxs) .= view(var, idxs)
+        continue
+      end
+
+      idxs_lhs, idxs, idxs_rhs = cellindices.(Ref(mesh), (k-1,k,k+1))
+      vvar_lhs, vvar, vvar_rhs = view.(Ref(var), (idxs_lhs,idxs,idxs_rhs))
+      vx_lhs,   vx,   vx_rhs   = view.(Ref(x),   (idxs_lhs,idxs,idxs_rhs))
+      vtmp_var                 = view(tmp_var, idxs)
+
+      # determine averages and slopes, including neighbors
+      v1, v2 = vvar_lhs[end-1], vvar_lhs[end]
+      x1, x2 = vx_lhs[end-1], vx_lhs[end]
+      avgs[1] = (v1+v2)/2; δs[1] = (v2-v1)/(x2-x1)
+      v1, v2 = vvar_rhs[1], vvar_rhs[2]
+      x1, x2 = vx_rhs[1], vx_rhs[2]
+      avgs[end] = (v1+v2)/2; δs[end] = (v2-v1)/(x2-x1)
+      for i in 1:Npts-1
+        v1, v2 = vvar[i], vvar[i+1]
+        x1, x2 = vx[i], vx[i+1]
+        avgs[1+i] = (v1+v2)/2; δs[1+i] = (v2-v1)/(x2-x1)
+      end
+
+      # limit slopes
+      for i in 2:Npts
+        # lim_δs[i-1] = TCI.minmod(δs[i-1],δs[i],δs[i+1])
+        x1, x2 = vx[i-1], vx[i]
+        dx = x2-x1
+        M = 0.1
+        δl, δc, δr = δs[i-1], δs[i], δs[i+1]
+        lim_δs[i-1] = TCI.minmod_M(dx, M, δc, δl, δr)
+      end
+
+      # recompute avgs so that continuity and total avg is preserved
+      fill!(A, 0.0) # because we use lu! below
+      for i in 1:Npts-2
+        x1, x2, x3 = vx[i], vx[i+1], vx[i+2]
+        dx1, dx2 = x2-x1, x3-x2
+        A[i,i]   = 1.0
+        A[i,i+1] = -1.0
+        b[i] = - 0.5*dx1*lim_δs[i] - 0.5*dx2*lim_δs[i+1]
+      end
+      A[end,:] .= 1.0
+      v_avgs = view(avgs, 2:Npts)
+      b[end] = sum(v_avgs)
+
+      # solve the equations
+      F = lu!(A)
+      ldiv!(lim_avgs, F, b)
+
+      # resample data
+      x1, x2 = vx[1], vx[2]
+      dx = x2-x1
+      vtmp_var[1] = lim_avgs[1] - 0.5*dx*lim_δs[1]
+      x1, x2 = vx[end-1], vx[end]
+      dx = x2-x1
+      vtmp_var[end] = lim_avgs[end] + 0.5*dx*lim_δs[end]
+      for i in 2:Npts-1
+        x1, x2 = vx[i], vx[i+1]
+        dx = x2-x1
+        vtmp_var[i] = lim_avgs[i] - 0.5*dx*lim_δs[i]
+      end
+    end
+
+    copyto!(var, tmp_var)
+  end
+end
+
+
+function limit_slope_convex_hull!(var::AbstractArray, mask::AbstractArray,
+                                   ws::Workspace, mesh::Mesh1d{<:DGElement})
+  enter(ws) do
+
+    @unpack x = get_static_variables(mesh)
+    Npts = mesh.element.Npts
+    K = mesh.tree.dims[1]
+    tmp_var = borrow(ws, Npts*K)
+    dvar = borrow(ws, Npts-1)
+    w = 0.5
+
+    for k in 1:K
+
+      idxs = cellindices(mesh, k)
+      vvar     = view(var, idxs)
+      vx       = view(x, idxs)
+      vtmp_var = view(tmp_var, idxs)
+
+      if k == 1 || k == K || mask[k] ≈ 0.0
+        # skip bdry cells, but backup data because we restore the whole vector at once later
+        vtmp_var .= vvar
+        continue
+      end
+
+      for i in 1:Npts-1
+        dvar[i] = vvar[i+1]-vvar[i]
+      end
+      if abs(sum(sign, dvar)) ≈ Float64(Npts-1)
+        vtmp_var .= vvar
+        continue
+      end
+
+      idxs_lhs, idxs_rhs = cellindices.(Ref(mesh), (k-1,k+1))
+      vvar_lhs, vvar_rhs = view.(Ref(var), (idxs_lhs,idxs_rhs))
+
+      vtmp_var[1] = vvar[1] + (vvar_lhs[end]-vvar[1])*w
+      vtmp_var[end] = vvar[end] + (vvar_rhs[1]-vvar[end])*w
+      for i in 2:Npts-1
+        ul, uc, ur = vvar[i-1], vvar[i], vvar[i+1]
+        xl, xc, xr = vx[i-1], vx[i], vx[i+1]
+        um = ul + (ur-ul)/(xr-xl)*(xc-xl)
+        vtmp_var[i] = uc + (um-uc)*w
+      end
+    end
+
+    copyto!(var, tmp_var)
+  end
+end
diff --git a/src/mesh.jl b/src/mesh.jl
index d400bfd9..9ef12398 100644
--- a/src/mesh.jl
+++ b/src/mesh.jl
@@ -920,59 +920,3 @@ function interpolate_face_data!(mesh::Mesh2d, var::AbstractVector, bdrybuf::Abst
     bo += Nptsx
   end
 end
-
-
-#######################################################################
-#                  slope limiters for FVElement mesh                  #
-#######################################################################
-
-
-function limit_slope!(δu, u, mesh::Mesh1d{<:FVElement}, @nospecialize(prms::NamedTuple))
-  @unpack hrsc = prms
-  if hrsc === :muscl
-    limit_slope_muscl!(δu, u, mesh, prms)
-  elseif hrsc === :fv_central
-    # TODO Not sure if correct. The discussion in Hesthaven book 2018, sec 10.3 does not
-    # specify how to compute the slope. The only requirement seems to be that
-    # the slope obeys δu_j/h = du/dx(x_j) + O(h). And in the codes they provide they
-    # seem to reuse the same slope limiters as for the MUSCL scheme etc.
-    prms = (hrsc = :muscl, muscl_omega=0)
-    limit_slope_muscl!(δu, u, mesh, prms)
-  else
-    TODO(hrsc)
-  end
-end
-
-
-function limit_slope_muscl!(δu, u, mesh::Mesh1d{<:FVElement}, prms::NamedTuple)
-  # assume that FV solution in jth cell [x_(j-1/2),x_(j+1/2)] can be approximated by
-  #   u_j ≈ ̄u_j + δu_j (x-x_j)/h
-  # where δu_j is the slope at x_j and
-  #   x_j = x_j-1/2 + h/2
-  #   h   = x_j+1/2 - x_j-1/2
-  # Compare discussion in Hesthaven book 2018, sec 10.2
-  @unpack muscl_omega = prms
-  ω = muscl_omega
-  # TODO Enabling upwinding in fv_muscl_burgers_sine (ω = 1.0) gives asymmetric results.
-  # Do we need to compute two slopes like in the slope limiter implementations in EulerEq etc?
-  K, = mesh.tree.dims
-  @turbo for k = 2:K-1
-    kâ‚‹, kâ‚Š = k-1, k+1
-    uâ‚‹, uâ‚€, uâ‚Š = u[kâ‚‹], u[k], u[kâ‚Š]
-    Δ⁻u = u₀-u₋
-    Δ⁺u = u₊-u₀
-    # (10.12) in Hesthaven book 2018, sec 10.2
-    δu[k] = (1+ω)/2*Δ⁻u + (1-ω)/2*Δ⁺u
-  end
-  if isperiodic(mesh)
-    for k in (1,K)
-      kâ‚‹, kâ‚Š = periodic_index(k-1,K), periodic_index(k+1,K)
-      uâ‚‹, uâ‚€, uâ‚Š = u[kâ‚‹], u[k], u[kâ‚Š]
-      Δ⁻u = u₀-u₋
-      Δ⁺u = u₊-u₀
-      δu[k] = (1+ω)/2*Δ⁻u + (1-ω)/2*Δ⁺u
-    end
-  else
-    TODO("bdry conditions")
-  end
-end
diff --git a/test/IntegrationTests/refs/burgers_sine_bspline1_limit_slope_convex_hull/burgers_sine_bspline1_limit_slope_convex_hull.toml b/test/IntegrationTests/refs/burgers_sine_bspline1_limit_slope_convex_hull/burgers_sine_bspline1_limit_slope_convex_hull.toml
new file mode 100644
index 00000000..fc3b95a3
--- /dev/null
+++ b/test/IntegrationTests/refs/burgers_sine_bspline1_limit_slope_convex_hull/burgers_sine_bspline1_limit_slope_convex_hull.toml
@@ -0,0 +1,23 @@
+[Evolution]
+cfl = 0.8
+tend = 0.24
+
+[ScalarEq]
+equation = "burgers"
+id = "sine"
+hrsc = "slope_limiter"
+
+[TCI]
+variables = ["u"]
+method = "mda"
+
+[Output]
+aligned_ts = "$(collect(range(0.02,0.24,step=0.02)))"
+variables1d = ["u"]
+
+[Mesh]
+dg_kind = "modal_bspline1"
+range = [0, 1]
+k = 13
+basis = "lgl"
+n = 7
diff --git a/test/IntegrationTests/refs/burgers_sine_bspline1_limit_slope_convex_hull/output1d.h5 b/test/IntegrationTests/refs/burgers_sine_bspline1_limit_slope_convex_hull/output1d.h5
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HcmV?d00001

diff --git a/test/IntegrationTests/refs/burgers_sine_bspline2_limit_slope_convex_hull/burgers_sine_bspline2_limit_slope_convex_hull.toml b/test/IntegrationTests/refs/burgers_sine_bspline2_limit_slope_convex_hull/burgers_sine_bspline2_limit_slope_convex_hull.toml
new file mode 100644
index 00000000..2849e536
--- /dev/null
+++ b/test/IntegrationTests/refs/burgers_sine_bspline2_limit_slope_convex_hull/burgers_sine_bspline2_limit_slope_convex_hull.toml
@@ -0,0 +1,23 @@
+[Evolution]
+cfl = 0.8
+tend = 0.24
+
+[ScalarEq]
+equation = "burgers"
+id = "sine"
+hrsc = "slope_limiter"
+
+[TCI]
+variables = ["u"]
+method = "mda"
+
+[Output]
+aligned_ts = "$(collect(range(0.02,0.24,step=0.02)))"
+variables1d = ["u"]
+
+[Mesh]
+dg_kind = "modal_bspline2"
+range = [0, 1]
+k = 14
+basis = "lgl"
+n = 7
diff --git a/test/IntegrationTests/refs/burgers_sine_bspline2_limit_slope_convex_hull/output1d.h5 b/test/IntegrationTests/refs/burgers_sine_bspline2_limit_slope_convex_hull/output1d.h5
new file mode 100644
index 0000000000000000000000000000000000000000..5e6c34bba2e750a0ed685d6e25bcb2ab679188c1
GIT binary patch
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diff --git a/test/IntegrationTests/refs/testconfig.toml b/test/IntegrationTests/refs/testconfig.toml
index fd03eac8..e9e6bbd4 100644
--- a/test/IntegrationTests/refs/testconfig.toml
+++ b/test/IntegrationTests/refs/testconfig.toml
@@ -61,6 +61,11 @@ variables1d = [ "u" ]
 # this test now runs into the discontinuity
 [burgers_sine_avmda]
 variables1d = [ "u" ]
+# bspline tests
+[burgers_sine_bspline1_limit_slope_convex_hull]
+variables1d = [ "u" ]
+[burgers_sine_bspline2_limit_slope_convex_hull]
+variables1d = [ "u" ]
 
 [balancedburgers_kink_avmda]
 variables1d = [ "u" ]
-- 
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