explicitly perform multiplications

lib/OrdinaryDiffEqFIRK/src/firk_perform_step.jl

@@ -1369,19 +1369,19 @@ end
 
     J = calc_J(integrator, cache)
     if u isa Number
+        LU1 = -γdt * mass_matrix + J
         tmp = -(αdt[1] + βdt[1] * im) * mass_matrix + J
     else
+        LU1 = lu(-γdt * mass_matrix + J)
         tmp = lu(-(αdt[1] + βdt[1] * im) * mass_matrix + J)
     end
     LU2 = Vector{typeof(tmp)}(undef,  (num_stages - 1) ÷ 2)
     LU2[1] = tmp
     if u isa Number
-        LU1 = -γdt * mass_matrix + J
         for i in 2 : (num_stages - 1) ÷ 2
             LU2[i] = -(αdt[i] + βdt[i] * im) * mass_matrix + J
         end
     else
-        LU1 = lu(-γdt * mass_matrix + J)
         for i in 2 : (num_stages - 1) ÷ 2
             LU2[i] = lu(-(αdt[i] + βdt[i] * im) * mass_matrix + J)
         end
@@ -1412,7 +1412,13 @@ end
             end
             z[i] = @.. z[i] * c_prime[i]
         end
-        w = TI*z
+        #w = TI*z
+        for i in 1:num_stages
+            w[i] = zero(u)
+            for j in 1:num_stages
+                w[i] += TI[i,j] * z[j]
+            end
+        end
     end
 
     # Newton iteration
@@ -1431,7 +1437,15 @@ end
         end
         integrator.stats.nf += num_stages
 
-        fw = TI * ff
+        #fw = TI * ff
+        fw = Vector{typeof(u)}(undef, num_stages)
+        for i in 1:num_stages
+            fw[i] = zero(u)
+            for j in 1:num_stages
+                fw[i] += TI[i,j] * ff[j]
+            end
+        end
+
         Mw = Vector{typeof(u)}(undef, num_stages)
         if mass_matrix isa UniformScaling # `UniformScaling` doesn't play nicely with broadcast
             for i in 1 : num_stages
@@ -1481,8 +1495,13 @@ end
         w = @.. w - dw
 
         # transform `w` to `z`
-        z = T * w
-
+        #z = T * w
+        for i in 1:num_stages
+            z[i] = zero(u)
+            for j in 1:num_stages
+                z[i] += T[i,j] * w[j]
+            end
+        end
         # check stopping criterion
         iter > 1 && (η = θ / (1 - θ))
         if η * ndw < κ && (iter > 1 || iszero(ndw) || !iszero(integrator.success_iter))
@@ -1565,7 +1584,6 @@ end
     mass_matrix = integrator.f.mass_matrix
 
     # precalculations
-
     γdt, αdt, βdt = γ / dt, α ./ dt, β ./ dt
     (new_jac = do_newJ(integrator, alg, cache, repeat_step)) &&
         (calc_J!(J, integrator, cache); cache.W_γdt = dt)
@@ -1600,7 +1618,13 @@ end
             end
             @.. z[i] *= c_prime[i]
         end
-        mul!(w, TI, z)
+        #mul!(w, TI, z)
+        for i in 1:num_stages
+            w[i] = zero(u)
+            for j in 1:num_stages
+                w[i] += TI[i,j] * z[j]
+            end
+        end
     end
 
     # Newton iteration
@@ -1620,13 +1644,19 @@ end
         end
         integrator.stats.nf += num_stages
 
-        mul!(fw, TI, ks)
+        #mul!(fw, TI, ks)
+        for i in 1:num_stages
+            fw[i] = zero(u)
+            for j in 1:num_stages
+                fw[i] += TI[i,j] * ks[j]
+            end
+        end
 
         if mass_matrix === I
             Mw = w
         elseif mass_matrix isa UniformScaling
             for i in 1 : num_stages
-                mul!(z[i], mass_matrix.λ, w[i])
+                mul!(z[i], mass_matrix.λ, w[i])       
             end
             Mw = z
         else
@@ -1698,7 +1728,13 @@ end
         end
 
         # transform `w` to `z`
-        mul!(z, T, w)
+        #mul!(z, T, w)
+        for i in 1:num_stages
+            z[i] = zero(u)
+            for j in 1:num_stages
+                z[i] += T[i,j] * w[j]
+            end
+        end
         # check stopping criterion
         iter > 1 && (η = θ / (1 - θ))
         if η * ndw < κ && (iter > 1 || iszero(ndw) || !iszero(integrator.success_iter))