What Mr. Erdogan needs is economic buttressing, and what Ms. Merkel needs is for Turkey to remain economically stable and for the Turkish leader to continue to stem the tide of refugees and migrants that pass through his country from war-torn nations — be they Syria, Iraq or Afghanistan — an issue that has become the chancellor’s Achilles’ heel.

Mr. Erdogan’s economy is teetering. The Turkish lira has come under heavy pressure — losing nearly 40 percent of its value this year — amid high inflation and spiraling foreign debt. Turkish private sector debt amounts to more than $200 billion, and as the lira has fallen in value Turkish companies have struggled to meet payments.

If he is not quite coming to Germany chastened, with cap in hand, Mr. Erdogan is certainly looking to reassure financial markets, improve business ties and seek political support.

It is a relatively steep climb-down for Mr. Erdogan. Relations with Germany hit rock-bottom last year with weeks of acrimonious sniping, as senior officials on both sides leveled accusations ranging from blackmail and hostage-taking to using Nazi tactics and abetting terrorism. Turkish newspapers branded German foundations and journalists as spies.

Exasperated with Mr. Erdogan’s jibes, and detentions of German citizens and threats to a list of its businesses in Turkey, Germany last summer announced a policy change that curtailed European economic assistance to Turkey.

Analysts say the policy has worked, though Turkish officials insist the mood has calmed because German and Turkish elections are now out of the way.