16.9.1. ERKStep Full RHS

digraph erk_fullrhs { node [shape=box] splines=ortho // ----------------- // Before first step // ----------------- init [label="Initialize integrator\lCopy t0, y0 to tn, yn\l"] f0cur [label="Set fn_is_current to False", style=filled, fillcolor=tomato1] h0 [label="Is h0 provided?", target="_top"] f0_q1 [label="Has f(tn, yn) been computed?"] eval1 [label="Evaluate f(tn, yn)\lStore in F[0]\lCopy F[0] to fn\l", style=filled, fillcolor=lightskyblue] fcur1 [label="Set fn_is_current to True", style=filled, fillcolor=lightgreen] h0_comp [label="Compute h0"] start [label="Start Step"] init -> f0cur -> h0 h0 -> start [taillabel="Yes", labeldistance=2, labelangle=45] h0 -> f0_q1 [taillabel="No", labeldistance=2, labelangle=-45] f0_q1 -> h0_comp [taillabel="Yes", labeldistance=2, labelangle=-45] f0_q1 -> eval1 [taillabel="No", labeldistance=2, labelangle=-45] eval1 -> fcur1 fcur1 -> h0_comp h0_comp -> start // ---------- // First step // ---------- f0q2 [label="Has f(tn, yn) been computed?"] step_q [label="Is this the first step after initial setup?"] eval2 [label="Evaluate f(tn, yn)\lStore in F[0]\lCopy F[0] to fn\l", style=filled, fillcolor=lightskyblue] fsal [label="Is the method FSAL?", style=filled, fillcolor=slateblue1] eval3 [label="Evaluate f(tn, yn)\lStore in F[0]\lCopy F[0] to fn\l", style=filled, fillcolor=slateblue1] copy1 [label="Copy F[s-1] to F[0]\lCopy F[0] to fn\l", style=filled, fillcolor=slateblue1] fcur2 [label="Set fn_is_current to True", style=filled, fillcolor=lightgreen] stages1 [label="Compute Stages"] complete [label="Complete Step"] start -> f0q2 f0q2 -> stages1 [taillabel="Yes", labeldistance=2, labelangle=45] f0q2 -> step_q [taillabel="No", labeldistance=2, labelangle=-45] step_q -> eval2 [taillabel="Yes", labeldistance=2, labelangle=45] eval2 -> fcur2 step_q -> fsal [taillabel="No", labeldistance=2, labelangle=-45] fsal -> eval3 [taillabel="No", labeldistance=2, labelangle=-45] eval3 -> fcur2 fsal -> copy1 [taillabel="Yes", labeldistance=2, labelangle=45] copy1 -> fcur2 fcur2 -> stages1 -> complete // ------------- // Complete step // ------------- interp_update [label="Interpolation enabled?"] interp_type1 [label="Using Hermite interpolation?"] f0q3 [label="Has f(tn, yn) been computed?"] update_yn [label="Copy tcur, ycur to tn, yn"] fcur3 [label="Set fn_is_current to False", style=filled, fillcolor=tomato1] complete -> interp_update interp_update -> update_yn [taillabel="No", labeldistance=2, labelangle=45] interp_update -> interp_type1 [taillabel="Yes", labeldistance=2, labelangle=-45] interp_type1 -> update_yn [taillabel="No", labeldistance=2, labelangle=45] interp_type1 -> f0q3 [taillabel="Yes", labeldistance=2, labelangle=-45] f0q3 -> update_yn [taillabel="Yes", labeldistance=2, labelangle=45] update_yn -> fcur3 // ---------- // After step // ---------- interp_eval [label="Evaluate interpolant?"] interp_type2 [label="Using Hermite interpolation?"] f1q1 [label="Has f(tn, yn) been computed?"] fsal1 [label="Is the method FSAL?", style=filled, fillcolor=slateblue1] eval4 [label="Evaluate f(tn, yn)\lStore in F[0]\lCopy F[0] to fn\l", style=filled, fillcolor=slateblue1] copy2 [label="Copy F[s-1] to F[0]\lCopy F[0] to fn\l", style=filled, fillcolor=slateblue1] fcur4 [label="Set fn_is_current to True", style=filled, fillcolor=lightgreen] interp_yout [label="Compute yout"] step2 [label="Start Next Step"] fcur3 -> interp_eval interp_eval -> step2 [taillabel="No", labeldistance=2, labelangle=45] interp_eval -> interp_type2 [taillabel="Yes", labeldistance=2, labelangle=-45] interp_type2 -> interp_yout [taillabel="No", labeldistance=2, labelangle=45] interp_type2 -> f1q1 [taillabel="Yes", labeldistance=2, labelangle=-45] f1q1 -> fsal1 [taillabel="No", labeldistance=2, labelangle=45] fsal1 -> eval4 [taillabel="No", labeldistance=2, labelangle=-45] eval4 -> fcur4 fsal1 -> copy2 [taillabel="Yes", labeldistance=2, labelangle=45] copy2 -> fcur4 fcur4 -> interp_yout -> step2 }

16.9.2. ARKStep Full RHS

digraph ark_fullrhs_start { node [shape=box, style=filled, fillcolor=white] splines=ortho bgcolor=lightskyblue // -------------- // Full RHS Start // -------------- fcur [label="Has f(tn, yn) been computed?"] eval [label="Evaluate fe(tn, yn), fi(tn,yn)\lStore in Fe[0], Fi[0]\l"] mass_a [label="Is there a mass matrix?"] mass_b [label="Is M time dependent?"] mass_c [label="Solve M(t) u = Fe[0], M(t) v = Fi[0]\lStore u, v in Fe[0], Fi[0]\lCopy Fe[0] + Fi[0] to f\l"] mass_d [label="Copy Fe[0] + Fi[0] to f\lSolve M x = f\lCopy x to f\l"] copy [label="Copy Fe[0] + Fi[0] to f"] return [label="Set fn_is_current to True", style=filled, fillcolor=lightgreen] fcur -> copy [taillabel="Yes", labeldistance=2, labelangle=45] fcur -> eval [taillabel="No", labeldistance=2, labelangle=45] eval -> mass_a mass_a -> copy [taillabel="No", labeldistance=2, labelangle=45] copy -> return mass_a -> mass_b [taillabel="Yes", labeldistance=2, labelangle=-45] mass_b -> mass_c [taillabel="Yes", labeldistance=2, labelangle=-45] mass_c -> return mass_b -> mass_d [taillabel="No", labeldistance=2, labelangle=45] mass_d -> return }

Fig. 16.1 ARKStep Full RHS Start

digraph ark_fullrhs_start { node [shape=box, style=filled, fillcolor=white] splines=ortho bgcolor=slateblue1 // ------------ // Full RHS End // ------------ fcur [label="Has f(tn, yn) been computed?"] sa [label="Is the method stiffly accurate?"] eval [label="Evaluate fe(tn, yn), fi(tn, yn)\lStore in Fe[0], Fi[0]\l"] mass_a1 [label="Is there a mass matrix?"] mass_a2 [label="Is there a mass matrix?"] mass_b1 [label="Is M time dependent?"] mass_b2 [label="Is M time dependent?"] mass_c [label="Solve M(t) u = Fe[0], M(t) v = Fi[0]\lStore u, v in Fe[0], Fi[0]\lCopy Fe[0] + Fi[0] to fn\l"] mass_d1 [label="Solve M x = fn\lCopy x to fn\l"] mass_d2 [label="Copy Fe[0] + Fi[0] to fn\lSolve M x = fn\lCopy x to fn\l"] copy_1 [label="Copy Fe[0] + Fi[0] to fn"] copy_2 [label="Copy Fe[s-1], Fi[s-1] to Fe[0], Fi[0]\lCopy Fe[0] + Fi[0] to fn\l"] return [label="Set fn_is_current to True", style=filled, fillcolor=lightgreen] fcur -> copy_1 [taillabel="Yes", labeldistance=2, labelangle=45] fcur -> sa [taillabel="No", labeldistance=2, labelangle=45] sa -> copy_2 [taillabel="Yes", labeldistance=2, labelangle=45] copy_2 -> mass_a1 mass_a1 -> return [taillabel="No", labeldistance=2, labelangle=-45] mass_a1 -> mass_b1 [taillabel="Yes", labeldistance=2, labelangle=45] mass_b1 -> mass_d1 [taillabel="No", labeldistance=2, labelangle=-45] mass_d1 -> return mass_b1 -> return [taillabel="Yes", labeldistance=2, labelangle=45] sa -> eval [taillabel="No", labeldistance=2, labelangle=-45] eval -> mass_a2 mass_a2 -> copy_1 [taillabel="No", labeldistance=2, labelangle=45] copy_1 -> return mass_a2 -> mass_b2 [taillabel="Yes", labeldistance=2, labelangle=-45] mass_b2 -> mass_c [taillabel="Yes", labeldistance=2, labelangle=-45] mass_c -> return mass_b2 -> mass_d2 [taillabel="No", labeldistance=2, labelangle=45] mass_d2 -> return }

Fig. 16.2 ARKStep Full RHS End

digraph ark_fullrhs { node [shape=box] splines=ortho // ----------------- // Before first step // ----------------- init [label="Initialize integrator\lCopy t0, y0 to tn, yn\l"] f0cur [label="Set fn_is_current to False", style=filled, fillcolor=tomato1] h0 [label="Is h0 provided?"] f0_q [label="Has f(tn, yn) been computed?"] rhs_1 [label="Call Full RHS Start", style=filled, fillcolor=lightskyblue] h0_comp [label="Compute h0"] start [label="Start Step"] init -> f0cur -> h0 h0 -> start [taillabel="Yes", labeldistance=2, labelangle=45] h0 -> f0_q [taillabel="No", labeldistance=2, labelangle=-45] f0_q -> h0_comp [taillabel="Yes", labeldistance=2, labelangle=45] f0_q -> rhs_1 [taillabel="No", labeldistance=2, labelangle=-45] rhs_1 -> h0_comp h0_comp -> start // ---------- // Start step // ---------- method_q [label="Is the first stage explicit?\nor\nIs the method stiffly accurate and Hermite interpolation is used?"] step_q [label="Is this the first step after initial setup?"] fn_q [label="Has f(tn, yn) been computed?"] rhs_2 [label="Call Full RHS Start", style=filled, fillcolor=lightskyblue] rhs_3 [label="Call Full RHS End", style=filled, fillcolor=slateblue1] stages [label="Compute Stages"] complete [label="Complete Step"] start -> method_q method_q -> stages [taillabel="No", labeldistance=2, labelangle=45] method_q -> fn_q [taillabel="Yes", labeldistance=2, labelangle=-45] fn_q -> stages [taillabel="Yes", labeldistance=2, labelangle=-45] fn_q -> step_q [taillabel="No", labeldistance=2, labelangle=-45] step_q -> rhs_2 [taillabel="Yes", labeldistance=2, labelangle=-45] step_q -> rhs_3 [taillabel="No", labeldistance=2, labelangle=-45] rhs_2 -> stages rhs_3 -> stages stages -> complete // ------------- // Complete step // ------------- interp_update [label="Interpolation enabled?"] interp_type1 [label="Using Hermite interpolation?"] f0q3 [label="Has f(tn, yn) been computed?"] rhs3 [label="Call Full RHS Start", style=filled, fillcolor=lightskyblue] update_yn [label="Copy tcur, ycur to tn, yn"] fcur1 [label="Set fn_is_current to False", style=filled, fillcolor=tomato1] complete -> interp_update interp_update -> update_yn [taillabel="No", labeldistance=2, labelangle=45] interp_update -> interp_type1 [taillabel="Yes", labeldistance=2, labelangle=-45] interp_type1 -> update_yn [taillabel="No", labeldistance=2, labelangle=45] interp_type1 -> f0q3 [taillabel="Yes", labeldistance=2, labelangle=-45] f0q3 -> update_yn [taillabel="Yes", labeldistance=2, labelangle=45] f0q3 -> rhs3 [taillabel="No", labeldistance=2, labelangle=-45] rhs3 -> update_yn -> fcur1 // ---------- // After step // ---------- interp_eval [label="Evaluate interpolant?"] interp_type2 [label="Using Hermite interpolation?"] f1q1 [label="Has f(tn, yn) been computed?"] rhs4 [label="Call Full RHS End", style=filled, fillcolor=slateblue1] interp_yout [label="Compute yout"] return [label="Start Next Step"] fcur1 -> interp_eval interp_eval -> return [taillabel="No", labeldistance=2, labelangle=45] interp_eval -> interp_type2 [taillabel="Yes", labeldistance=2, labelangle=-45] interp_type2 -> interp_yout [taillabel="No", labeldistance=2, labelangle=45] interp_type2 -> f1q1 [taillabel="Yes", labeldistance=2, labelangle=-45] f1q1 -> interp_yout [taillabel="Yes", labeldistance=2, labelangle=45] f1q1 -> rhs4 [taillabel="No", labeldistance=2, labelangle=45] rhs4 -> interp_yout -> return }