--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+rv/+time/BDFDerivative.m	Wed Dec 05 15:04:44 2018 +0100
@@ -0,0 +1,33 @@
+class BDFDerivative < handle
+    properties
+        dt
+        coefficents
+    end
+    methods
+        function obj = BDFDerivative(order, dt)
+            obj.coefficents = obj.getBDFCoefficients(order);
+            obj.dt = dt;
+        end
+
+        function DvDt = evaluate(v, v_prev)
+            DvDt = (obj.coefficients(1)*v - sum(obj.coefficients(2:end).*v_prev),2)/obj.dt;
+        end
+
+        function c = getBDFCoefficients(obj, order)
+            switch order
+                case 1
+                    c(1,1) = 1; c(1,2) = 1;
+                case 2 
+                    c(1,1) = 3/2; c(1,2) = 4/2; c(1,3) = -1/2;
+                case 3
+                    c(1,1) = 11/6; c(1,2) = 18/6; c(1,3) = -9/6; c(1,4) = 2/6;
+                case 4
+                    c(1,1) = 25/12; c(1,2) = 48/12; c(1,3) = -36/12; c(1,4) = 16/12; c(1,5) = -3/12;
+                case 5
+                    c(1,1) = 137/60; c(1,2) = 300/60; c(1,3) = -300/60; c(1,4) = 200/60; c(1,5) = -75/60; c(1,6) = 12/60;
+                case 6
+                    c(1,1) = 147/60; c(1,2) = 360/60; c(1,3) = -450/60; c(1,4) = 400/60; c(1,5) = -225/60; c(1,6) = 72/60; c(1,7) = -10/60;
+            assert(length(c) == (order+1));
+        end
+    end
+end

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+rv/+time/RungekuttaExteriorRV.m	Wed Dec 05 15:04:44 2018 +0100
@@ -0,0 +1,57 @@
+classdef RungekuttaExteriorRV < time.Timestepper
+    properties
+        F       % RHS of the ODE
+        k       % Time step
+        t       % Time point
+        v       % Solution vector
+        n       % Time level
+        coeffs  % The coefficents used for the RK time integration
+        RV      % Residual Viscosity
+        v_prev  % Solution vector at previous time levels, used for the RV update
+        DvDt    % Function for computing the time deriative used for the RV update
+    end
+    methods
+
+        function obj = RungekuttaExteriorRV(F, k, t0, v0, RV, rkOrder, bdfOrder)
+            obj.F = F;
+            obj.k = k;
+            obj.t = t0;
+            obj.v = v0;
+            obj.n = 0;
+            % Extract the coefficients for the specified rkOrder
+            % used for the RK updates from the Butcher tableua.
+            [s,a,b,c] = time.rk.butcherTableau(rkOrder);
+            obj.coeffs = struct('s',s,'a',a,'b',b,'c',c);
+        
+            obj.RV = RV;
+            % TODO: Initialize v_prev.
+            obj.v_prev = zeros(length(obj.v),bdfOrder);
+            %  TBD: For cases where h~k we could probably use rkOrder-2 here.
+            obj.DvDt = rv.time.BDFDerivative(bdfOrder,k);
+        end
+
+        function [v, t] = getV(obj)
+            v = obj.v;
+            t = obj.t;
+        end
+
+        function state = getState(obj)
+            [residual, u_t, grad_f] = obj.RV.getResidual();
+            state = struct('v', obj.v, 'residual', residual, 'u_t', u_t, 'grad_f', grad_f, 'viscosity', obj.RV.getViscosity(), 't', obj.t);
+        end
+
+        function obj = step(obj)
+            % Store current time level    
+            obj.v_prev(:,2:end) = obj.v_prev(:,1:end-1);
+            obj.v_prev(:,1) = obj.v;       
+            % Fix the viscosity of the RHS function F
+            F_visc = @(v,t)F(v,t,obj.RV.getViscosity()); %Remove state in RV?
+            obj.v = time.rk.rungekutta(obj.v, obj.t, obj.k, F_visc, obj.RV, obj.coeffs);
+            % Calculate dvdt and update RV
+            dvdt = obj.DvDt.evaluate(v, v_prev, lenght(v_prev));
+            RV.update(v,dvdt);
+            obj.t = obj.t + obj.k;
+            obj.n = obj.n + 1;
+        end
+    end
+end
\ No newline at end of file